1
∆
36 Zg wewmGm cixÿv_© x‡`i Rb¨ MwYZ wel‡qi ¸iæZ¡c~ c~Y©Y© `y w !"¨vq #web¨vm $ mgv‡e%& wb‡q we'( vwiZ !v‡)v*bv +iv ,)- gb‡.vM w`‡q /03 2v c‡) wewmGm G cvi‡eb Ge4 cieZx© ‡Z 5. 5+vb cixÿvq !vm‡) wb‡R w b‡R wb‡R cvi‡eb .viv gvbwe+ we7vM we 7vM 5_‡+ cv‡)8v c v‡)8v +‡i‡9b Zv‡`i Zv ‡`i +v‡9 G !"¨vq `y ‡v !‡b+ +w:b +w :b g‡b ,q w+;< ey ‡= ey ‡= +i‡) !‡b+ m,‡R cviv m>? e- G8v‡b Ggb7v‡e 5)8v ,‡q‡9 .v‡Z c~ e© eZ© x 5+vb !vw@qv 9vv wb‡R wb‡R mgv"vb +i‡Z cv‡ib !"¨vq `y wi ABm ,) #!vgvi 5)8v # Khairul’s Basic Math” ewi wCZxq m4'Di‡Y !v‡iv we'( vwiZ !v‡)v*bvEEmgv"vbFFg‡@) 5 m, m4‡.vR‡bi +vR *)‡9- mg‡qi mg‡q i mHZvi mHZv i +vi‡b cI_g _g m4'DiY w G+J w@ w@ +‡i mevi mv,v.¨v‡_© w`‡q w`‡q w`)vg wb‡R w%8y b !b¨‡+ w%8‡Z m,‡.vMxZv +iæbmg‡qi 'KHZvi H Zvi +vi‡Y me‡_‡+ ¸iæZ¡c~ cY©Y~ © welq L w7w@$ MZix +iv ,q wb- cieZ© cieZ© x‡Z w7w@$ MZix +‡i mevi mv‡_ 5%qvi +i‡ev- mv‡_6 _v+Jbb-
web¨vm OPermutationP wewmGm Gi bZJ b wm‡)ev‡m bZJ b7v‡e G !"¨vqw .y N ,$qvq !‡b‡+i G !"¨vqw 7v‡)v7v‡e ey =‡Z mgm¨v ,qGLv‡b Ly e mnRfv‡e ev¯Í eZvi mv‡_ wgwj‡q GB Aa¨vqwU Ggbfv‡e Av‡jvPbv Kiv n‡q‡Q, † †K! †"# $% &Í ey ey ‡' ey ‡' $(‡j Av"v Kwi wb‡R †_‡KB web¨vm m)*v&Í me me $+ ‡" i !-i w.‡Z $vi‡eb/
OPermutationP w+Q KZ0wj e¯1 †_‡K K‡qKwU ev meKwU A_ev wbw.% wbw.% 2 K‡qKwU $+ wZev‡i wb‡q Z fv‡e mvRv‡bv vq Zv‡.i $+ ‡Z¨KwU‡K GK GKwU web¨vm e‡j/ !.vni34 g‡b Kwi A, B, C, wZbwU e3%/ GKmv‡_ meKwU e3% wb‡q wb‡q mvRv‡bv vq/ ABC, ACB, BAC, BCA, CAB, CBA ‡gvU 5 fv‡e/ v‡.i $+ wZwU‡K GK GKwU web¨vm e‡j/ my Ziv) !$‡iv6 !.vni3 †_‡K ey 'v vq meKwU 7UbvB GK GKwU web¨vm ev mvRv‡bvi e¨e¯8 v Zvn‡j †gvU mvRv‡bvi m vRv‡bvi e¨e¯8 v n‡jv 5 wU/ !.vni34 g‡b Kwi A,B,C wZbwU e3%/ GKmv‡_ .y BwU e3% K‡i K‡i wb‡q mvRv‡bv vq/ AB, BA, AC, CA, BC, CB/
∆
/
ev'( ‡e cI ‡qvML
Qv9:Qv9;‡.i ‡ivj b<= i, >v(;i jvB‡m?, †gvevBj b<= i, †fvUvi AvBw@ Kv‡@% i b<= i †_‡K $% &Í C wU w@wRU wb‡qB †KvwU †KvwU m)L¨v evbv‡bv nq, vi GKwU mv‡_ Ab¨ ‡KvbwUi wgj †bB/
web¨v‡mi my L n m)L¨K wewfb e¯1 n‡Z $+ wZev‡i r m)L¨K e¯1 wb‡q †gvU mvRv‡bvi e¨e¯8 v
n!
n
pr =
(n - r )!
K_vq 4 npr =
!$‡ iiUv D¨‡ ± vwiqvj ( !$‡ ii : wb‡ Pi ) D¨v‡ KUvwiqvj
my ‡9i
e¨vL¨v4 GLv‡b n! A_% n‡jv n Gi mv‡_ Zvi wb‡Pi mKj *wgK m)L¨vi 03Dj/ †gb4 awi n Gi gvb 5 Ge) r Gi gvb 2/ Zvn‡j gvb0‡jv ewm‡q my 9wU wb‡< v6 wbq‡g e¨venvi Ki‡Z n‡e, 5
p2=
5!
(5 - 2)!
E 5 ! E 5 × 4 × 3 × 2 × 1 E20 A_ev 5 ! E 5 × 4 × 3! E GLb !$‡ii F wb‡Pi 3! †K ‡K‡U 3 × 2 ×1
3!
3!
3!
w.‡j ay 5×4 _v‡K E20
weL`SL G‡H‡9 g‡b iv8‡Z ,‡e bv¸w) cy biveT wU ,‡e bv Ge4 n Ge4 r !e%¨ !v)v`v gvb e,b +i‡eFactorial +x $ 5+bQ Factorial (!) n‡IQ †Kvb abvZJK $3 m)L¨vi % 03b wewa v C K‡i K‡g *gv&= ‡q 0b n‡q C $% &Í n‡e/ †gb, M E ×C, NM E N××C, OM E O× N××C BZ¨vw./ Permutation F Combination G 03b wewaB mgm¨v mgvav‡bi †gwjK †K"j/ GUv‡Z C K‡i K‡g vq †.Lv vK, A,B,C ‡_‡K NwU !$v.vb wb‡q web¨vm Ki‡Z $+_‡g A, B, C wZbwUB †bqv vq/ wK&1 $‡i CwU option K‡g vq/
GLv‡b, $+_‡g A, B, C wZbwU option ‡bqv vq/ $‡i A Gi Rb¨ .y wU B,C), B Gi Rb¨ (C,A), C Gi Rb¨ A,B) Gfv‡e .y wU K‡i option _v‡K/ ZvB N× Gi $i Avi GKwU K‡i gv9 option _v‡K/ †Kbbv, AB Gi Rb¨ C Qv(v Avi †K! †bB/ BC Gi Rb¨ A Qv(v Avi †bB/ Kvi3, N e‡3% i .y wU‡Zv G‡mB †>‡Q/ GRb¨ N××C = 5wU Permutation web¨vmS m
V!vgiv G !4‡% wew7bW "i‡bi Permutation wb‡q !v‡)v*bv +i‡evL .8b me Acv`vb w7bW Lb me !$v.vb wfb ZLb Permutation .y wU we#‡qi !$i wbf% i K‡i/ C Gi !$v.vb m)L¨v F KZ !$v.vb wb‡Z n‡e/ G‡H‡9 !$v.vb m)L¨v n( ‡gvU !$v.vb‡K n Vviv $+ Kv" Kiv nq ) Ge) r m)L¨K !$v.vb wb‡Z n‡j, web¨vm m)L¨v nPr v e¨vL¨v K‡i .vW (vq n, C K‡i K‡g r av$ $% &Í / Avgiv Permutation F Combination .yX‡H‡9B av$ †K gyL¨ we‡eP¨ we#q ivL‡ev v mgm¨v mgvavb A‡bK mnR Ki‡e/ GKwU !.vni3 †.qv vK/ :
1F /F 3F XF G *viw m48¨v 5_‡+ X !4‡+i +Z¸w) m48¨v M:b m>? eQ G‡H‡9 †‡nZY m)L¨v O A)‡Ki O !$v.vb wb‡Z n‡e e‡jS †m‡nZY G‡.i av‡$ av‡$ O av$ $% &Í mvRv‡Z n‡e/ GLv‡b v n‡e, OpO E OM ev O × N × × C
3
∆
wK&1 w. e‡j A‡i KZ0wj m)L¨v >[b m
.8b w+9J Acv`vb w7bWL Avgiv GZH‡b Rvwb, n m)L¨K wfb !$v.vb wb‡R‡.i g‡a¨ n! !$v‡q web¨¯Í nq/ wU wfb !$v.v‡bi Rb¨ ×C, NwU wfb !$v.v‡bi Rb¨ N ××C:::: Gfv‡e/ Kvi3, $+_g av‡$i $i $+ wZ av‡$ CwU K‡i option Kg‡Z _v‡K/ .8b cy biveT wU ‡L mvavi3Z Permutation G repetition allowed bq/ wK&1 wKQY mgm¨vi ‡H‡9 repetition n‡Z $v‡i/ G‡H‡9 awi, n m)L¨K wfb e¯1 †_‡K r m)L¨K ¯8 vb $ i3 Ki‡Z n‡e/ G‡H‡9 Cg ¯8 vb n $+ Kv‡i $ i3 Kiv v‡e/ Kvi3 repetition Pj‡e/ Gfv‡e Nq ¯8 vbwUF n $+ Kv‡i $ i3 Kiv v‡e/ ∴ r m)L¨K ¯8 vb nr $+ Kv‡i $ i3 Kiv v‡e/ †gb, $ bive\ w- K‡i A, B, C wZbwU !$v.vb †_‡K Kqfv‡e wU !$v.vb †bqv v‡e GLv‡b, mKj web¨vm n‡e Gi]$, AA, AB, AC, BA, BB, BC, CA, CB, CC, 9 wU/ †Kbbv $+ wZ †H‡9B $+ wZ av‡$ Av‡>i me options ‡_‡K vq/ r 2 GK e3% wiw$U Kiv †>‡j Gfv‡e/ ∴ G‡H‡9 web¨vm m)L¨v n = 3 = 9/ A_% v^
weL`SL GLv‡b $y bive\ w- 7‡UbvB ZvB GBm 9 $+ ‡qv> Kiv n‡q‡Q/ Vc~YiveT wUi m~ L n m)L¨K e¯1 i meKwU
G‡Kev‡i wb‡q web¨vm m)L¨v wb3% q Ki‡Z n‡e Lb p m)L¨K e¯1 GK $+ Kvi q m)L¨K wVZ;q $+ Kvi r m)L¨K Z Z;q $+ Kvi Ge) evK; e¯10wj wfb wfb ‡‡nZY †gvU mvRv‡bvi m)L¨v n‡e E
n! p!q!r!
cI %WL web¨vm m48¨v wbY© q +iQ E2, 3, 4, 5, 6, 7, $ 8 G !¸w)i cI ‡Z¨+w‡+ cI ‡Z¨‡+i m48¨vq G+evi gv e¨e,vi +‡i *vi !‡i +Z¸w) cT_+ m48¨v M:b +iv 5.‡Z cv‡i mgvavb4 ‡gvU m)L¨v n = 7 , wb‡Z n‡e 4 wU K‡i i e r = 4 ‡gvU mvRv‡bvi e¨e¯8 v n
pr =
n!
(n - r )!
Avevi, npr =
=
7!
(7 - 4 )!
n!
=
=
7!
=
3!
8!
(n - r )! (8 - 3)!
=
7 × 4 × 5 × 4 × 3!
8! 5!
3!
E 7×6×5×4
= 840 (ans):
E 8 × 7 × 6 × 5! E 336 5!
c\wZ]1L mv"viY web¨vm wKQY $+ " Av‡Q †0‡jv web¨v‡mi my 9 Qv(vB gy ‡L gy ‡L Kiv vq/ †gb4
1E %v,vevM 5_‡+ [vg© ‡M‡i wZbw w7bW iv'( v !v‡9F !vevi [vg© ‡M 5_‡+ ebvbxi Xw w7bW iv'( v !v‡9- %v,evM 5_‡+ [vg© ‡M ,‡q ebvbx .vevi +qw w7bW iv'( v !v‡9Q
∆
X
mgv"vbL N×O E CwU `e¨v8¨vL Kvi3 $+_gevi A‡a% K iv¯Í v GK $v" w.‡q w>‡q evK; A‡a% K Ab¨ $v‡" w.‡q †>‡j
Gfv‡e †gvU wfb wfb iv¯Í vi m)L¨v n‡e CwU
/E ^v+v ,‡Z ewi%v) .vevi c_ 3w Ge4 ewi%v) ,‡Z 8y )bv .vevi c_ w ,‡)F +Z Acv‡q ^v+v ,‡Z 8y )bvq .v$qv .v‡eQ mgv"vbL vKv n‡Z Ly jbvq †‡Z $vi‡e E N × c ECc 3EG+w 5%I wY+‡ÿ 3w `iRv !v‡9- +Z7v‡e G+Rb w%ÿ+ +‡ÿ ^J +‡Z $ 5ei ,‡Z cvi‡eb e¨v8¨vL †‡nZY wZbwU w.‡q Y K‡e Zv‡nj d wZwbwUi wfb wfb wU w.‡q †ei n‡Z $vi‡e Ge) †vKvi mgq F wfb wfb .iRv w.‡q y K‡e/ ZvB †gvU N× N E fv‡e Y K‡Z F †ei n‡Z $vi‡eb/ XEG+w 5%I wY+‡ÿ 3w `iRv !v‡9- +Z7v‡e G+Rb w%ÿ+ G+ `iRv w`‡q ^J ‡+ !b¨ `iRv w`‡q 5ei ,‡Z cv‡ibQ 5 e¨v8¨ v4 GLv‡b †‡nZY Ab¨ .iRv w.‡q †ei nFqvi K_v ejv n‡q‡Q ZvB † .iRv w.‡q Y K‡e †m .iRv w.‡q †ei nFqv v‡e bv/ GKwU A$"b K‡g vFqvq !-iwU n‡e N × E 5wU/ E Rvex‡ii 3w w7bW i‡i c¨v2 $ 3w w7bW i‡i %v© !v‡9- 5m +Z7v‡e G+ 5Rvv c¨v2]%v© c92` +i‡Z cv‡iQ 6E ^v+v $ wm‡) iæ‡ cI wZw`b w 5cab *)v*) +‡i- AN `y w ' v‡b .vZvqvZ +iv .v‡e +Z Acv‡qQ e¨vL¨v .y wU ¯8 v‡b vZv‡qZ Kivi !$vq E cc:CS E cO E !$v‡q Ans.20 Kvi3 † †$ ‡b v‡e †m ‡$ ‡b Avm‡e bv/ Ge) †gvU ‡$ b cwU/ ZvB GKwU K‡g 03 Kiv n‡q‡Q
c\wZ]f/L wew7bW %‡i web¨vm wbqg]1L mv"vib %‡i web¨vm O‡.8v‡b 5+vb e‡Y© i cyYiveT wU bv _v+‡)P 1En m48¨+ wew7bW e'< ,‡Z cI wZev‡i n m48¨+ e'< d!_v©B G+mv‡_ me+w wb‡q 5gv mvRv‡bvi e¨e' v my 94
n!
n
pr =
(n - r )!
GB m 9wU e¨envi Ki‡eb/E n ! 0!
=
n! 1
= n! we4.h4 0! = 1 A_% v^ M E C nq/
!.vni34 16EA, B, C cI wZev‡i `y w eY© G+mv‡_ wb‡q +Z cI +v‡i mvRv‡bv .vq wbY© q +i- dcy biveT wU bv w‡q mgv"vbL GLv‡b †gvU e3% n = 3 .y BwU K‡i e3%G Kmv‡_ wb‡Z n‡e/ †gb4 r = 2 n
pr =
n!
(n - r )!
=
3!
(3 - 2)!
=
3! 1!
=
3 × 2 ×1 1
[ 3! = 3 × 2×1 = 6 ]
+mv‡_ wb‡q +Z cI +v‡i mvRv‡bv .vq wbY© q +i17.A, B, C cI wZev‡i wZbw +‡i eY©G
mgv"vbL †gvU e3% n = 3
∆
wZbwU e3%G Kmv‡_ wb‡Z n‡e †gb4 n = 3 n
pr =
n!
=
(n - n )!
3!
=
(3 - 3)!
3!
=
0!
3 × 2 ×1
=6
1
Shortcut: Z0‡jv e3% _vK‡e w. me0‡jv wb‡Z e‡j Zvn‡j ‡Kvb my 9 bv wj‡L mivmwi d e3% 0‡jvi
†gvU m)L¨vi mv‡_ M ev D¨v‡wiqvj †v> K‡i w.‡q 03 Ki‡jB n‡q v‡e/ †gb4
1ETable, %w‡+ +Z7v‡e mvRv‡bv .vqQ Gfv‡e ej‡j †Kvb "Z% bv _vK‡j ey '‡Z n‡e me0‡jv e3% wb‡Z n‡e mgv"vbL mivmwi 5! = 5×4×3×2 = 120 1hE A, B, C, D *viw eY© - eY *viw © ,‡Z 5+vb c~ biveT wU bv w‡q wZbw +‡i eY© wb‡q mvRv‡bvi e¨e' v wbY© q +i mgv"vbL †gvU e3% n = 4 $+ wZev‡i wb‡Z n‡e 3 wU e3% % r = 3 (GLv‡b KZwU e3% wb‡Z n‡e Zv e‡j †.qv Av‡Q ) ‡gvU mvRv‡bvi e¨e¯8 v n
pr =
n!
(n - r )!
=
4!
(4 - 3)!
=
4! 1!
=
4 × 3 × 2 ×1 1
= 24
/fE DAUGHTER "wU w.‡q KZ0‡jv wfb wfb " >[b K‡i vq Zv wb3% q Kikb4
mgv"vbL ‡gvU e3 n% = 8 wb‡Z n‡e meKwU = 8 { Kvi3 GLv‡b ejv nqwb KZwU e3% wb‡Z n‡e } ‡gvU mvRv‡bvi e¨e¯8 v n
pr=
n!
(n - n )!
E
8!
(8 - 8)!
E 8 ! E 8! 0!
1
E 8×7×6×5×4×3×2×1= 40320
m)‡H‡$ 8! = 40320
wb‡R +iæbL /1EcI wZw Letter G+evi e¨e,vi +‡i ORANGE ‡_‡+ +Z¸w) !ÿ‡ii % M:b +iv .vqQ /f d6p5 //EcI wZw Letter G+evi e¨e,vi +‡i Versity %w‡+ +Z7v‡e mvRv‡bv .v‡eQ
w+9J ¸iæZ¡ ©c~ [¨v‡wiqv) m48¨vi gvb gy8'( iv8‡Z cv‡ibL 5.gbL 0! = 1,
1! = 1,
2! = 2
3! = 6,
4! = 24,
5!= 120,
6! = 720, 7!= 5040
wbqg]/L ‡+vb Acv`vb G+vw"+evi _v+‡)F !_v© B cyYiveT wU ,‡)F n m)L¨K !$v.vb GKB n‡j
G‡.i wb‡q gv9 CwU web¨vm nq/ Kvi3 $+ wZ av‡$B CwU gv9 option _v‡K/ †gb , , wb‡q wZb A‡i GKwUB A >[b m
my L
n! p!×q!×r!
GLv‡b n n‡jv †gvU !$v.vb m)L¨v, Avi p, q, r, n‡jv GKvwaKevi e¨elZ !$v.vb m)L¨vS
6
∆
wKQY !.vni3 †.‡L Av‡iv wK qvi nFqv v‡e/ wb‡Pi "0‡jvi web¨vm m)L¨v †ei Kikb/ CS AMERICA 2) CANADA 3) CALCUTTA g‡b ivL‡eb, GB m 9 ZLbB e¨envi Ki‡eb Lb †Kvb m)L¨v ev AHi G‡Ki AwaK evi Ae¯8 vb Ki‡e/
mgv"vbL 1P AMERICA %w‡Z 5gv 7 w eY© !v‡9 .vi g‡"¨ A !v‡9 2 w Ge4 ev+x¸‡)v 'K Z;j wb‡Y© q web¨vm m48¨v E 7!
=
2!
7 × 6 × 5 × 4 × 3× 2 ×1 2 ×1
= 42 ×20×3 = 42× 60 = 2520 (ans)
2) CANADA %w‡Z 5gv !ÿi !v‡9 6 w Gi g‡"¨ A !v‡9 3 w my Ziv) wb‡3% q web¨vm m)L¨v E 6 ! E 6 × 5 × 4 × 3 × 2 × 1 = 120 3! 3× 2 ×1 3) CALCUTTA %w‡Z 5gv eY© !v‡9 8 w .vi g‡"¨ A !v‡9 2 wF C !v‡9 2 w Ge4 T !v‡9 2 w 8!
!ZGe web¨vm m48¨v E E E
2!2!2! 8 × 7 × 6 × 5 × 4 × 3 × 2 ×1 2 × 1× 2 × 1× 2 × 1 8× 7 × 6× 5× 4×3 4
= 8×7×6×5×4×3 = 56×90 = 5040 (ans) 1.CALCUTTA %wi eY© ¸‡)v‡+ G+‡ wb‡q m48¨vi +Z ¸YQ `35 Zg wewmGm K KS 2 >S 4 LS 3 7S 5
web¨vm m48¨v
AMERICA %wi eY© ¸‡)v G+‡ wb‡q web¨vm
mgv"vbL AMERICA "wU‡Z 7 wU e3% Av‡Q v‡.i g‡a¨ wU A i‡q‡Q/ me0‡jv e3% GK‡9 wb‡q web¨vm m)L¨v E CALCUTTA "wU‡Z wU
m)L¨v E =
2
7! 2!
8! 2!2!2! =
=
8 × 7! 2!2!2!
=
7! 2!
e3% Av‡Q v‡.i g‡a¨ A, C F T 2 wU K‡i we.¨gvb/ me0‡jv e3% GK‡9 wb‡q web¨vm 8 × 7! 2 × 2 × 2!
‡jv GK‡9 wb‡q web¨vm E K 2 × AMERICA "wUi e3%0
A_ev .y wUi Avjv.v Avjv.v !-i †ei K‡i e(wU‡K †QvUwU Vviv fv> w.‡jF !-i GKB Avm‡e/
wb‡R +iæbL 5gv +Z7v‡e web¨'( +iv .vqQ 21.1E #LEADER & %‡i eY© ¸w)‡+ 5gv +Z7v‡e web¨'( +iv .vqQ 22.# DHAKA & %‡i eY© ¸w)‡+ 23.# RAJSHAHI & %‡i eY© ¸w)‡+ 5gv +Z7v‡e web¨'( +iv .vqQ
36f
∆
24. me¸w) eY© G+evi wb‡q committee %w‡+ +Z i+‡g mvRv‡bv .vqQ X36f 5gv +Z7v‡e web¨'( +iv .vqQ 25.# Book & %‡i eY© ¸w)‡+
wbqg]3L wbw`© k 5+vb Acv`vb wbw`© k ' v‡b !_ev cv%vcvw% iv8‡Z e)‡) wbw`© k ' v‡b iv8‡Z e)‡)F /6E Apu %wi eY© ¸‡)v wb‡q +Z¸‡)v web¨vm m48¨v wbY© q +iv .vq 5.b cI ‡Z¨+ web¨v‡mi cI_‡g G+w Vowel _v‡+QO mgv"vbL GLv‡b Apu "wU‡Z †gvU wZbwU e3% Av‡Q/ Avevi †KvbwUB .y evi ‡bB/ ZvB G‡.i web¨vm m)L¨v n‡e/ NM ev N× E 5wU/ GLb ev¯Í ‡e fvey b/ wZbwU AHi w.‡q w. †gvU 5wU web¨vm Kiv vq/ Zvn‡j $+ wZ C wU w.‡q 5÷ N E wU K‡i web¨vm mvRv‡bv vq/ Avevi Apu "wU‡Z †‡nZY .y wU ¯= ie3% _v4 A F U w.‡q †gvU web¨vm n‡e × E OwU/ ‡gb4 Apu, Aup, upa, uap ZvB !-i4 O/ wK&1 GB $+ " wU‡ZB w. ejv nZ $+_‡g Vowel _vK‡e bv GiKg web¨vm m)L¨v KqwU n‡e ZLb !-i nZ wU ‡ .y wU P w.‡q ik nq/ †gb4 pau, pua.
wb‡R +iæbL /E Cap %wi eY© ¸‡)v wb‡q +Z¸‡)v web¨vm m48¨v wbY© q +iv .vq 5.b cI ‡Z¨+ web¨v‡mi cI_‡g G+w 'K ieY© _v‡+ /ENokia’ %wi eY© ¸‡)v wb‡q +Z¸‡)v web¨vm m48¨v wbY© q +iv .vq .v‡`i iæ‡Z Consonant _v‡+Q !‡ovS Nokia "wU‡Z †gvU cwU e3% Av‡Q ZvB cM E C fv‡e mvRv‡bv vq/ Zvn‡j c AH‡ii $+ wZwU w.‡q mvRv‡bv v‡e C÷c E OwU K‡i/ GLb Nokia "wU‡Z †‡nZY .y wU Consonant Av‡Q, ZvB Consonant w.‡q ik n‡e †gvU O× E OwU web¨vm/ !-i4 OwU/ e( e( web¨vm Avm‡jF wbqg GKBS
wb‡R +iæbL
/hE Courage’ %wi eY© ¸‡)v wb‡q +Z¸‡)v web¨vm m48¨v wbY© q +iv .vq 5.b cI ‡Z¨+ web¨v‡mi cI_‡g G+w 'K ieY© _v‡+Q = L KS Cp LS >S N5O 7S †KvbwUB bq ‡gvU web¨vm ‡ei K‡i pw.‡q fv> w.‡q Ow.‡q 03S
`S æZ [¨v‡wiqv) Gi +vv+w +ivi Rb¨ wb‡*i A`v,iY 5`8y b6! 3!
E 6 × 5 × 4 × 3 × 2 × 1 Gfv‡e $+ wZwU m)L¨v wjL‡Z A‡bK mgq jv>‡e ZvB .h kZ KvUvKwU Ki‡ 3 × 2 ×1
Gfv‡e wj‡L KvUvKvwU Kikb
6!
3! wb‡Pi 3!
E
6 × 5 × 4 × 3! 3!
`wb‡Pi 3! ‡K bv †f‡r ‡i‡L w.b/ Ge) !$‡ii 6! †
$% &Í fvry b/ GLb Ge) !$‡ii 3! †K‡U w.‡q Aew"20‡jv 0b Kikb/ mgq evW P‡e/ E120 B vm‡e/ Gfv‡e !$‡i wb‡P e( e( D¨v‡wiqvj Avm‡j GKwU wgwj‡q †K‡U w.‡q Ab¨0‡jv 03 Kikb/
3!
cv%vcvw% iv8‡Z e)‡)F 3fESCIENCE %wi 'K ieb© ¸‡)v‡+ G+‡ 5i‡8 me +qw eY© © ‡+ m>? e¨ .Z Acv‡q mvRv‡bv .vq Zvi m48¨v wbY© q +i`KZ0‡jv e3%G K‡9 †i‡L A_ev $\_K bv †i‡L Kivi wbqg CS viv GK‡9 _vK‡e Zv‡.i‡K GKwU e3 we‡ePbv % K‡i $+ v e3% ev e¯1 ‡K web¨vm Ki‡Z n‡e/
∆
S vivGK‡9 _vK‡e bv Zv‡.i web¨v¯Í Ki‡Z n‡e/ CS F S b) Gi $+ v web¨vm m)L¨vi 0bDj n‡e wb‡3% q web¨vm m)L¨v p wU mgv"vbL SCIENCE "wU‡Z e3% m)L¨v ¯= ie3% wZbwU‡K GK‡9 †i‡L GKwU e3% we‡ePbv Ki‡j A_% v^ SCNC (EEI) e3% m)L¨v n‡e cwU ¯= ie3%0 ‡jv‡K GK‡9 ivL‡Z ejv n‡q‡Q, ZvB Giv ‡Lv‡bB emy GK mv‡_ em‡e/S ∴SCNC (EEI) Gi mvRv‡bvi m)L¨v E †gvU cwU e3% †Lv‡b .y wU C ♦
Kvi3 $+ ‡"
E 5 ! E 5 × 4 × 3 × 2 × 1 E 60 2!
2 ×1
Avevi (EEI) Gi web¨vm m)L¨v E 3 ! 2!
`‡‡nZY E = 2 wU E 3 × 2 × 1 E 2 ×1
3
{ y w6,
wZbwU ¯= ie3% $v"v$vw"
ivL‡jB nj, ZvB GB ¯= ie3% 0‡jv EEI , EIE , A_ev IEE, Gfv‡e Avm‡jF "Z% $ i3 n‡e/ GRb¨ .y evi web¨vm Ki‡Z nj/ } AZGe wb‡3% q web¨vm m)L¨v E 60 × 3 = 180
wb‡R +iæbL
31EVowel ¸w) G+mv‡_ 5i‡8 +Z7v‡e Problem %w web¨vm +iv .v‡e ` Help: Prblm(oe) = 6! ‡"‡# 2! 03 COO 3/E Vowel ¸w) G+mv‡_ 5i‡8 Acclaim %w‡+ +Z7v‡e mvRv‡bv .v‡eQ `Help: cclm (aai) =5! evK;Uv wb‡R Kikb C 33EParallel %wi Vowel ¸w)‡+ G+‡ 5i‡8 5gv web¨vm m48¨v +Z ,‡eQ N5 3XEVowel ¸w) cv%vcvw% bv 5i‡8 Triangle’ %wi !ÿi¸‡)v +Z7v‡e mvRv‡bv .v‡eQ 36fff mgv"vbL Triangle "wUi ¯= ie3% NwU iae ‡K GKwU e3% a‡i, evK; cwU consonant mn 5wU AHi trngl (iae) a‡i 5M $+ Kv‡i ev p fv‡e mvRv‡bv vq/ Avevi ay Vowel 0‡jv‡K mvRv‡bv vq NM iK‡g ev N× E 5 fv‡e / Zvn‡j vowel 0‡jv‡K $v"v$vw" †i‡L †gvU mvRv‡bvi m)L¨v n‡e p×5 E ON iK‡g / ∴ G‡H‡9 †gvU web¨vm m)L¨v nj ON/ GUv nj $v"v$vw" †i‡L !-i4 wK&1 $+ ‡" ejv n‡q‡Q $v"v$vw" bv ‡i‡L ZvB Avevi, Triangle G wU wfb e3% Av‡Q v‡.i M E × p × 5 × c × O × N × × E ON iK‡g mvRv‡bv vq/ GB ON Gi g‡a¨ $v"v$vw" †i‡L , bv †i‡L me ai‡3i web¨vm Av‡Q/ GLvb †_‡K ey ‡' wbb † Triangle "wU‡K †gvU ON fv‡e mvRv‡bv vq, vi ga¨ †_‡K Vowel 0‡jv‡K $v"v$vw" ‡i‡L †gvU mvRv‡bvi m)L¨v ON/ Zvn‡j Vowel 0‡jv‡K $v"v$vw" bv †i‡L mvRv‡bvi m)L¨v n‡e ON : ON E N5/ !-i4 N5 3EMathematics %wi eY© ¸w)‡+ mvwR‡q G‡`i +Zw‡Z Vowel G+‡ _v+‡eQ C5
c\wZ]f3L wew7bW m48¨vi web¨vm weE`SL Ac‡ii wbqg¸w)i gZ- cy biveT wU ‡) cy biveT wUi m~ cI ‡qvM +‡i +i‡Z ,‡eF !vi cy biveT wU bv ‡) mv"viY wbq‡g +i‡Z ,‡e-
∆
h
36E 1F /F 3F XF F 6 !¸‡)v cI wZw Gevi wb‡q X !‡i +Z¸w) w7bW m48¨v ,‡eQ 36f mgv"vbL mvavi3 wbq‡gi gZB nPr = 5PO = = 5M = 5 × c × O × N × = 360 5 − OSM M
wb‡R +iæbL 3EcI ‡Z¨+w !4+ 5+e) G+evi +‡i e¨e,vi +‡i 1F/F3 Cviv +Z¸‡)v `y !4+ wew%k m48¨v M:b +iv .vqQ 3
P/ E5wU
3E N, N, O, O, O, c m)L¨v0‡jv w.‡q 5 Awew"2 KZ0‡jv wfb m)L¨v >[b Kiv v‡e 5 mgvavb4 N, N, O, O, c, c m)L¨v0wj‡Z K‡i K‡i N, O, c Av‡Q/ ∴G‡H‡9
web¨vm m)L¨v E
n! p!q!r!
` GLv‡b n n‡IQ †gvU Ge)
p, q
Ge) r n‡IQ $y bive\ - m)L¨v
5M 5 × c × O × N × E = M×M×M ×× 3hE 3F 3F XF XF F m48¨v¸w) w`‡q 6 !‡i +Z¸w) w7bW m48¨v M:b +iv .vqQ XfE1F /F 3F XF !¸w)i cI wZw‡+ 5. 5+vb m48¨+ evi wb‡q 3 !‡i +Z¸w) m48¨v M:b +iv .v‡eQ Cc mgv"vbL † †Kvb m)L¨K evi A_% $+ wZeviB C wb‡q N A)‡Ki m)L¨v n‡e CCC A_v%^ GLv‡b wiw$U Kiv v‡e/ =
my 94 GiKg $y bive\ w- Kiv †>‡j †gvU !$v.vbS Zevi †bqv v‡e GLv‡b †gvU !$v.vb C,,N,O F c E cwU Ge) m)L¨v evbv‡Z n‡e N A)‡Ki / ZvB !-i n‡e c N E CcwU †gb4 CN,CC,CCC, GiKg C w.‡q cwU mn $+ wZwU m)L¨v w.‡q c wU K‡i †gvU Cc wUS Av‡iKwU †.Ly b4 X1EF F 6F 6F F m48¨v¸w) 5_‡+ 3 !4‡+i +Z¸w) m48¨v M:b +iv .v‡eQ O
Vmgv"vbL GLv‡b † †Kvb m)L¨v BIQvgZ †bqv v‡e bv / Z‡e $+ ‡" $+ .- m)L¨v0‡jv‡KB ay e¨envi Kiv v‡e, †Lv‡b †gvU 5wU m)L¨v †.qv _vK‡jF Avm‡j m)L¨v !$.vbS †gvU NwU Ge) $+ wZwU evi K‡i Av‡Q/ ZvB Avgiv wjL‡Z $vwi, NwU !$v.vb w.‡q NA)‡Ki m)L¨v evbv‡Z n‡j wjL‡Z n‡e N N E pwU / wK&1 jH¨ Kikb NwU !$.vb wK&1 B‡IQ gZ wiw$U Kiv v‡e bv/ ei) † KqwU m)L¨v †.qv Av‡Q Zv †_‡KB wb‡Z n‡e/ GLv‡b c, 5 Ge) p Av‡Q wU K‡i/ wK&1 pwU m)L¨vi g‡a¨ Ggb NwU m)L¨v Av‡Q †Lv‡b ccc, 555, Ge) ppp Av‡Q †0‡jv †bqv v‡e bv/ Kvi3 $+ ‡" NwU K‡i m)L¨v †.qv †bB / ZvB GB NwU ev. w.‡j †gvU m)L¨v n‡e p:N E OwU / X/EcI wZw ! G+evi e¨e,vi +‡i XF 3F /F 1F f !¸w) Cviv !‡i +Z¸w) we‡Rv m48¨v M:b +iv .v‡eQ N5 Vmgv"vbL GLv‡b cwU m)L¨v Av‡Q Ge) $+ wZwU C evi K‡i wb‡Z n‡e,, ZvB cwU m)L¨v †K mvRv‡bv vq cM ev C fv‡e/ v‡.i g‡a¨ $+ wZwU m)L¨v w.‡q ik F †"# n‡e C÷c E OwU GLv‡b "Z% .y wU4 C m)L¨v0‡jv c A)K wew"2 n‡Z n‡e m)L¨v0‡jv †e‡Rv( n‡Z n‡e/ C F N w.‡q †"# n‡Z n‡eS ‡gvU ‡e‡Rv( m)L¨v4 † m)L¨v0‡jvi †"‡# C F N _vK‡e †m0‡jvB †ev‡Rv( ZvB ‡gvU †e‡Rv( OsO E OwU wK&1 GB O wU †e‡Rv( m)L¨vi g‡a¨ Ggb wKQY m)L¨v Av‡Q †0‡jv w.‡q ik n‡q‡Q v †ev‡Rv( wK&1 c A)K wew"2 bq/ ZvB Vviv i †e‡Rv( m)L¨v0‡jv ev. w.‡Z n‡e/
∆
1f
GLb w.‡q i nFqv †gvU m)L¨v C ÷c E OwU / Vviv † m)L¨v0‡jv ik n‡q‡Q †m0‡jv †"# n‡e C, , N, A_ev O w.‡q/ A_v^ O ÷O E 5 $+ wZwU w.‡q 5wUS Ge) C F N w.‡q †"# nFqv m)L¨v0‡jv we‡Rv( ZvB †gvU †e‡Rv( 5s5 E CwU/ GLb GB CwU m)L¨v †gvU †e‡Rv( m)L¨v O †_‡K ev. w.‡Z n‡e ZvB !-i4 O:C E N5wU / !-i4 N5wU/ t0ikZu$3 % w"Hb;q we#q4 E Vviv †Kvb m)L¨v ik nq bv/ E ‡Rv( A_ev †e‡Rv( †ei Ki‡Z ejv n‡j †"‡#i m)L¨v †gjv‡Z n‡e/ E †gvU web¨vm m)L¨v‡K !$v.vb m)L¨v Vviv fv> Ki‡j $+ wZwU !$v.vb w.‡q KZwU web¨vm ik F †"# n‡e Zv †ei n‡q/
c\wZ]fXL e\ -vKv‡i mvRv‡bv Note: eT Uv+v‡i mvRv‡Z e)‡)F
!v‡Mi mv"viY wbq‡gi gZ- my w cI ‡qvM +i‡Z ,‡eFw+;< .8b eT Uv+v‡i emv‡bv ,‡e Z8b 5%‡li 5)v+w cI_g 5)v+wi mv‡_ wg‡) .v‡e- Zv .Z Rb‡+ eT Uv+v‡i mvRv‡Z e)v ,‡e ZZ w`‡q n AUiw‡+ 7vM +i‡Z ,‡e5.gbL X3E +Z7v‡e 4 Rb 5)v+ G+w eT Uv+vi 5we‡)i *vicv‡% em‡Z cv‡iQ 6 n
pr 4
E
4
p4 4
E
4! 4
=
4 × 3× 2 4
=
6
XXE 6 R‡bi `) 5_‡+ +Z7v‡e 5 Rb eT Uv+vi7v‡e `vo v‡Z cv‡iQ 144 `Help:
6
p5 5
1. g‡@) 5
c~Y gvbL © 1f
O!"¨vqL web¨vmP
1EMobile %wi eY© gv)v¸‡)v‡+ +Z7v‡e mvRv‡bv .vqQ KCc Lp >C 7COO /ESUCCESS %‡i me eY© wb‡q +Zw w7bW w7bW % M:b +iv .v‡eQ KCc Lp >O 7OO 3EVowel ¸w) G+mv‡_ 5i‡8 Rajshahi %w‡+ +Z7v‡e mvRv‡bv .v‡eQ OScience Gi gZP KCc Lp >C 7COO XEVowel ¸w) G+mv‡_ 5i‡8 +Z7v‡e trouble %w web¨vm +iv .v‡eQ KCc Lp >C 7COO E 3F 3F XF XF XF m48¨v¸‡)v w`‡q 6 !wew%k +Z¸‡)v w7bW m48¨v M:b +iv .v‡eQ Kc L > 75 6E 3F 3F XF XF F m48¨v¸w) w`‡q 6 !‡i +Z¸w) w7bW m48¨v M:b +iv .vqQ K LC > 75
mgqL 1fwgwb
∆
11
E1F /F 3F XF F 6 !¸‡)v cI wZw G+evi wb‡q X !‡i +Z¸w) w7bW m48¨v ,‡eQ KCc Lp >C 7N5 E 6 R‡bi `) 5_‡+ +Z7v‡e 4 Rb eT Uv+vi7v‡e G+w 5Mv)‡we) pe:+ +i‡Z cvi‡e Kc L >CO 7p hESCIENCE %wi cI ‡Z¨+w eY© w`‡q +Z Acv‡q %w‡+ mvRv‡bv .v‡eQ KS C5 LS C5 >S C5 7S †KvbwUB bq 1fECourage’ %wi eY© ¸‡)v wb‡q +Z¸‡)v web¨vm m48¨v wbY© q +iv .vq 5.b cI ‡Z¨+ web¨v‡mi cI_‡g G+w 'K ieY© LS >S N5O 7S †KvbwUB bq _v‡+Q KS Cp
AUigv)vL
g‡@) 5]f1Oweb¨vmP 1E 8
/E M
3E M
XE 8
E
6E +
E
E 8
hE +
1fE 8
mgv‡e% OCombination) mgv‡e% OCombination) w+Q mgv‡e" n‡jv K‡qKwU !$v.vb †_‡K $+ ‡Z¨Kevi wbw.% 2 wKQY !$v.vb wb‡q GK GKwU .j >[b Kiv/ GLv‡b avivevwnKZv $wieZ% b n‡jF .‡ji m)L¨v GKB _vK‡e/ n!
mgv‡e‡"i my 94nCr=
r! × (n
− r)!
( web¨v‡mi
my ‡9i gZB ay AwZwi6 wn‡m‡e n‡ii mv‡_
r! 03 Ki‡Z n‡e/
web¨vm ebvg mgv‡e% OPermutation Vs Combination) Combination Gi †H‡9 Order ("vivevw,+Zv ) †Kvb Factor bq/ 0ikZu$3 % Ge) Order Gi $wieZ% b n‡j m)L¨viF $wieZ% b n‡e/ ‡gb4
wK&1
Permutation
Gi †H‡9 avivevwnKZv
wewfb $i;Hvi $+ ‡" Lb GB .y wU Aa¨vq †_‡K $+ " Avm‡e ZLb wj‡L †.qv _vK‡e bv †KvbwU web¨m Ge) †KvbwU mgv‡e" n‡e/ fv‡jvfv‡e $v_% K¨ bv Rvb‡j GKUvi Rvq>vq Ab¨wUi !-i K‡i †Dj‡Z $v‡ib/ ZvB G‡.i ga¨Kvi $v_% K¨0‡jv wb‡P QK AvKv‡i ZY ‡j aiv nj/
web¨vm $ mgv‡e‡%i g"¨+vi 5gw)+ cv_©+¨ O8y e ¸iæZ¡c~Y©P web¨vm O PermutationP
mgv‡e% OCombinationP
web¨vm n‡jv K‡qKwU !$v.vb †_‡K $+ ‡Z¨Kevi wbw.% 2 K‡qKwU wb‡q mvRv‡bvi m)L¨v / †Lv‡b avivevwnKZvi $wieZ% b n‡j bZY b web¨vm Zi; nq/ !.vni34AB ‡K web¨¯Í Kiv vq AB, BA, GB 2 fv‡e, G‡H‡9 avivevwnKZv $wieZ% b n‡j web¨vm m)L¨v $wieZ% b n‡e/
mgv‡e" n‡jv K‡qKwU !$v.vb †_‡K $+ ‡Z¨Kevi wbw.% 2 wKQY !$v.vb wb‡q GK GKwU .j >[b Kiv/ GLv‡b avivevwnKZv $wieZ% b n‡jF .‡ji m)L¨v GKB _vK‡e/ wK&1 AB Gi †K wb‡q GKevi gv9 mgv‡e" Kiv vq Zv nj, AB, AB .y Rb gvby # n‡j .y R‡b wg‡j GKwU .y, m.m¨ wew"2 .j/ avivevwnKZv $wieZ% b Ki‡jF .y m.‡m¨i GKwU .jB _vK‡e/
1/
∆
Av‡iKwU !.vni34ABC ‡K web¨¯Í Kiv vq AB, BA, AC,CA,BC, F CB Gik$ 5 fv‡e / G‡H‡9 avivevwnKZv $wieZ% b nFqvq web¨vm m)L¨v $wieZ% b n‡e/ web¨v‡mi my 9 nPr= n! (n
−
r)!
wK&1 ABC, Gi ga¨ †_‡K Rb K‡i wb‡q mgv‡e" Kiv v‡e AB, AC, Ge) BC GB wZbwU >+ k‡$/ wK&1 G‡H‡Z AB, ‡K BA evbv‡j bZY b mgv‡e" Avm‡e bv/ †Kbbv .y ‡Uv GKB we#q/ mgv‡e‡"i my 9 n‡jv nCr= n!
evg $v‡" AwZwi6 r! ‡bBS
r! × (n − r)!
evg $v‡"i AwZwi6
r!
wU
0ikZu$3S% web¨v‡mi !-i e( nq mgv‡e‡"i !-i †QvU nq/ mvavi3Z wewfb e3% w.‡q mvRv‡bvi A)K0‡jv †gb mvavi3Z wewfb KwgwU, ‡Ljvi wUg, Kig.% b, †Ljvi DHAKA, Ge) wewfb m)L¨vi † A)K 0‡jv Avm‡e ‡gvU m)L¨v †ei Kivi ‡H‡9 mgv‡e‡"i my 9 $+ ‡qv> †gb4 ,N,O,c w.‡q m)L¨v >[3, G0‡jv web¨v‡mi Ki‡Z n‡e/ †Kbbv G‡H‡9 avivevwnKZv $wieZ% b my 9vby vq; Ki‡Z n‡e/ †Kbbv, G0‡jvi †H‡9 n‡jF m)L¨vi $wieZ% b n‡e bv/ †gb4 wg A , wg B avivevwnKZvi $wieZ% b n‡jB GK GKwU bZY b web¨vm Gi mv‡_ Kig.% b Kiv gv‡bB wg B ,wg A Gi mv‡_ Zi; n‡e/ A_% v^ c †K 7y wi‡q wjL‡j c nq, v bZY b Kig.% b Kiv/ †jvK .y Rb‡K †w.‡KB †.b GKwUB web¨vm/ Kig.% b n‡e/ mgv‡e‡"i †H‡9 ev)jv‡.": fviZ Avi fviZ :ev)jv‡." Gi †Ljv A_% .y wU †Ljv bv ei) GKwU †Ljv/
web¨vm mgv‡e‡%i g‡"¨ 5.b A‡)v cv‡)v 5)‡M bv .vq Zv wb‡* A`v,iY w`‡q ‡ev=v‡bv ,‡)v- gb‡.vM w`‡q cr b-
8y e ¸iæZ¡c~Y© G+w ev'( em>sZ A`v,iYL ev4)v‡`%F 7viZ $ cvw+'( vb G+`‡)i mv‡_ !v‡i+`) G+w +‡i g¨v* 58)‡) 5gv +Zw 58)v ,‡e 3C2 = NwU ‡gb4 C ev4)v‡`% ebvg 7viZ ev4)v‡`% ebvg cvw+'( vb N 7viZ ebvg cvw+'( vb♦
♦ Avevi
ev)jv‡.", fviZ, ;j)Kv "+ F $vwK¯Í vb GK.j Av‡iK.‡ji mv‡_ GKwU K‡i g¨vP †Lj‡j †gvU g¨v‡Pi m)L¨v ! n‡e C2= !(!!) E ! ! E E 6 wU `memgq wb‡P em‡e Kvi3 GKwU g¨v‡P .y wU .j Ave"¨K 4
×
×
×
♣
cI gvb 5`8y bL
1E ev4)v‡`% ebvg 7viZ /E ev4)v‡`% ebvg x)4+v %I 3E ev4)v‡`% ebvg cvw+'( vb XE 7viZ ebvg x)4+v %I E 7viZ ebvg cvw+'( vb 6E x)4+v %I ebvg cvw+'( vb
×
×
×
×
×
∆
13
wew7bW c\wZi mgv‡e‡%i cI %W c\wZ] f1L `) M:b $ +ig`© b GB $xwZ‡Z Avgvi w"L‡ev wKfv‡e K‡qKRb †L‡jvqv‡ii †fZi †_‡K KZfv‡e GKwU w*‡KU, DY Uej,ev‡¯Uej A_ev † ‡Kvb .j >[b Kiv v/ mv‡_ mv‡_ wKfv‡e Ge) KZfv‡e GKwU .‡ji AwabvqK A_ev mn AwabvqK wbe% vwPZ Kiv vq/ .j >[‡bi mgq wewfb ‡L‡jvqv‡ii bvg Av‡> A_ev $‡i LbB ejv †nvK bv †Kb Zviv GKwU .jB †ev'v‡e, ZvB .j >[‡bi A)K 0‡jv mgv‡e‡"i my 9vby vq; Ki‡Z nq/
1E1f Rb 5)v+ cI ‡Z¨‡+ cI ‡Z¨‡i mv‡_ +ig`© b +‡i- +ig`© b m48¨v +ZQ X mgvavb4 † †Kvb Kig.%3 A_ev †KvjvKY wji A)‡K ay KZ Rb †jvK Kig.% b HandshakeS ev †KvjvKY wj Ki‡jv Zv †.qv _vK‡e/ G‡H‡9 g‡b ivL‡Z n‡e † $+ ‡Z¨K evi Kig.% b ev †KvjvKY wj Kivi mgq †gvU Rb †jv‡Ki $+ ‡qvRb/ ZvB G‡H‡9 ymy 9wU n‡e nCr E ‡gvU ‡jvK
CRb me mgq
cI `U cI %W wi mgv"vbL 10C2 =
10! 2!(10 − 2)!
=
10! 2!8!
=
10 × 9 × 8! 2 × 8!
=
10 × 9 2
= 5×9 =
45 Ans:45
Shortcut Solution: G ai‡bi A)K LvZv Kjg Qv(vB gy ‡L gy ‡L Kikb Gfv‡e Z
bB †.qv _vK Zvi Av‡>i m)L¨vi mv‡_ d m)L¨vwU 03 K‡i w.‡q fv> Ki‡jB !-i †ei ‡q v‡e/ wb‡Pi 0‡jv K‡iB †.Ly b, K‡qKwU Ki‡j wb‡RB ey '‡Z $vi‡eb/
wb‡R +iæbL
/E 3 Rb e¨wN 5_‡+ +Z7v‡e / R‡bi `) M:b +iv 5.‡Z cv‡iQ 3C2 N 3E 6 Rb 5)v+ G+ Rb !v‡i+R‡bi mv‡_ +ig`© b +i‡) 5gv +ig`© ‡Yi m48¨v +ZQ O Help: 5×cEN÷ E CcwUS XEG+w t`Mv, gq`v‡b 5gv 1ff Rb gy mw)a w9)- bvgv. 5%‡l cI ‡Z¨‡+ cI ‡Z¨‡+i mv‡_ 5+v)v+J w) +i‡) ‡gv +Zw ‡+v)v+J w) m4w:Z ,‡qw9%)Q Oc n¨vz‡"‡Ki m)L¨v †.qv _vK‡e, †gvU ‡jv‡Ki m)L¨v †ei Ki‡Z ejv n‡j !‡ovfv‡e Avm‡j4 A_% v^ ‡gb4
EG+w [Zvi cvw© ‡Z +‡q+Rb euJ Acw' Z ,)- Zviv cI ‡Z¨‡+ cI ‡Z¨‡+i mv‡_ +ig`© b +i)- .w` 5gv / w +ig`© b ,‡q _v‡+ Zv,‡) n cvw© ‡Z 5gv +ZRb Acw' Z w9)Q
mgv"vbL
!$‡ii wbqgwUi B !‡ov $+ ‡qv> Kikb Gfv‡e ‡f‡e †f‡e mevi †"‡# w.‡q fv> Kivq n‡q‡Q Zvn‡j fv> Kivi Av‡> wQj × E c5/ GLb GB c5 †K Ggb fv‡e fvr‡Z n‡e v‡Z $i $i .y wU m)L¨v 03 Ki‡j c5 nq/ m)L¨v .y wU nj ×p E c5/ GLb e( m)L¨vwU B n‡jv wb‡3% q !$w¯8 Z †jv‡Ki m)L¨v/
wb‡R +iæbL
6EG+w cvw© ‡Z w+9J 5)v+ Acw' Z w9)- Zviv cI ‡Z¨‡+ cI ‡Z¨‡+i mv‡_ ,¨v‡%+ +ivq ‡gv 66 w ,¨v‡%+ ,‡)v- n cvw© ‡Z 5gv +Z Rb Acw' Z w9)QQ C Help: 66×2 = 132 = 12×11S !$‡ii GB wbq‡gi $+ wZwU A)K !‡ov$v‡" †f‡e †f‡e mgvavb Ki‡j mn‡R fY ‡j v‡eb bv/
∆
1X
E/f m`m¨wew%k G+w [J e) `) ,‡Z G+Rb !w"bvq+ $ G+Rb m,]!w"bvq+ +Z7v‡e wbe© v*b +iv .v‡eQ C †‡nZY R‡bi †_‡K AwabvqK F mn:AwabvqK A_% v^ †gvU Rb wb‡Z n‡e, ZvB my 9wU n‡e4 20!
20
C2=
=
20 × 19 × 18!
2! ( 20 − 2 )!
=
20 × 19
2!18!
2
= 10 × 9 =
90
%© +v© L G-wKQY bv wj‡L Ly e mn‡R Ki‡Z PvB‡j Rb †_‡K .y Rb wb‡Z n‡e ZvB F Zvi Av‡>i m)L¨v 03 K‡i w.‡q fv>
w.‡jB †ei n‡q v‡e, gy ‡L gy ‡L K‡qK †m‡K‡z/ wK&1 Av‡> ey ‡' wbb Zvi$i Gfv‡e ‡P2v Kikb/ E 14 Rb 58‡)vqv‡i g‡"¨ 5_‡+ wbw`© k G+Rb !w"bvq+m, 11 R‡bi G+w w‡+ `) +Z7v‡e ev9v +iv .v‡eQ `35 Zg wewmGm L KS 728 >S 364 LS 286 7S 1001
mgv"vbL
‡‡nZY AwabvqK‡K ev. †.qv v‡e bv, ZvB AwabvqK memgq wD{U ev wbw.% 2/ evK; CN R‡bi ga¨ †_‡K C Rb evQvB Ki‡Z n‡e/ GKRb‡K AwabvqK wn‡m‡e evQvB Kivi $i evwK (14 - 1) ev 13 Rb n‡Z (11 - 1) ev 10 Rb‡K evQvB K‡i 11 R‡bi .j >[‡bi !$vq E 13C10
=
13!
=
13 × 12 × 11 × 10!
10!(13 − 10 )!
10!×3!
!vevi GZ m48¨v bv w)‡8 mivmwi G7v‡e w)8v .vq
=
13 × 12 × 11 3 × 2 ×1
13 × 12 × 11 3 × 2 ×1
=
=
286
286 +viY
13 5_‡+ 1f !_© wZbw +g Zv Ac‡i 13 Gi
5_‡+ 1 +‡i +g wZbw ivw% w)8vi ci wb‡* mgv‡e‡%i Rb¨ !wZwiN 3! w`‡q 7vM +i‡Z ,‡e-
4‡iRx‡Z !vm‡)$ wbqg G+ "y !_© ey ‡= +i‡Z ,‡e- 5.gbL If there are 20 boys , how many different basketball teams could be formed? there are 5 boys needed in each team. Answer : 15,504 Solution: !
Here, nCr =
!()!
Here n = 20 and r = 5
So we can calculate it now by putting the value 20
C5 =
!
=
×××××!
××××
! ×!
!()!
=
= 15,504
Solve youtselves:
hEIf 20 boys go out for football team, how many different teams may be formed, one at a time? = 167,960 [ Help: Here the Combination is
20
C11=167960 ]
1fEIn how many ways can a coach choose frist a football team and then a basketball team if 18 boys go out for either team? = 668 & 340
c\wZ]f/L +wgw ev m`m¨ M:b ‡+b wew7bW +wgw M:‡bi !4+¸‡)v mgv‡e‡%i my vby .vqx +i‡Z ,qQ G+w +wgw‡Z 3 Rb m`m¨ _v+vi !_© 3 m`m¨ wew%k +wgw- G8b 5.7v‡e .v‡+ !v‡M !_ev c‡i w`bF +wgw G+ _v+‡e- 5.gbL
∆
1
1fE 6 Rb ev)+ $ X Rb evw)+v 5_‡+ m`m¨wew%k +wgw +ivi +qw c_ !v‡9 5.8v‡b w:+ / Rb evw)+v .v‡eQ 1/f
mgv"vbL
evwjKv Rb _vK‡j N Rb evjK _vK‡e/ Zvn‡j O Rb evwjKv †_‡K Rb, Ge) 5 Rb evjK †_‡K N Rb wb‡q KwgwUi $+ Kvi E 4
C2
6
C3 =
wb‡R +iæbL
4×3 2 ×1
×
6× 5× 4 3 × 2 ×1
= 120.
w7bW w7bW cI +v‡i 5bqvi ci ¸Y +i‡Z ,q
11EG+w +z v‡ei Rb cy iæl $ Rb gw,)v m`m¨ !v‡9b- 6 m`‡m¨i G+w +wgw M:b +i‡Z ,‡e- 5.8v‡b cy iæl $ gw,)v m`m¨ 3 Rb +‡i _v+‡eb- +Z7v‡e G +wgw M:b +iv .vqQ NCN5 d Help: 8C3 8C3 = 3136] 1/E In how many ways can a subcommittee of 4 peosons be formed from amongst 4 women & 6 men so that are particular man is always included? a. 126 b. 504 c. 3024 d. 210 = b
Solution: 5 Rb $y ik# †_‡K C Rb wb‡ZB n‡e/ Aew"2 Rb †_‡K evK; N Rb wb‡Z n‡e 9×8×7 6 9 ∴ !$KwgwU $+ Kvi C1 × C3 = 6 × d hR‡bi 5_‡+ 3Rb Zv we‡qvM +‡i wZbw Acv`vb 5bq ,‡q‡9 3 × 2 ×1 = 9 × 8 × 7 = 504 !-i4 cO fv‡e/
13EG+w +z v‡ei Rb m`m¨ !v‡9- +z vew .w` X R‡bi +wgw M:b +i‡Z *vqF Z‡e +Zw w7bW w7bW +wgw M:b +iv .v‡eQ f 1XE Rb cy iæl $ 6 Rb gw,)v 5_‡+ R‡bi G+w +wgw wbe© v*b +i‡Z ,‡e- +wgw‡Z !;( ZL 3 Rb cy iæl _v+‡)F +Z7v‡e Gw +iv .v‡eQ 6 mgv"vbL A&Í Z ejvq ey '‡Z n‡e GLv‡b NRb $y ik# B wbw.% 2 bq/ ei) me% wb< N Rb, A_ev O, A_ev c RbF $y ik# _vK‡Z $v‡i/ ZvB A)KwU Ki‡Z n‡e Gfv‡e $+_‡g c R‡bi ga¨ ‡_‡K Avjv.v Avjv.v fv‡e Ns A_ev OsC A_ev cs †ei K‡i me0‡jv †v> K‡i w.‡Z n‡e/
1EG+Rb cixÿv_x© ‡+ 1/ w cI %W ,‡Z 6 wi AUi +i‡Z ,‡e- cI _g wi w:+ Xw cI %W ev9v +‡i +Z cI +v‡i 6w cI %W AUi +iv .v‡eQ Cc 16E 6w w7bW i‡i cZv+vi G+w ev G+vw"+w G+evi wb‡q +Zw m4‡+Z 5`qv .v‡eQ Cc5wU CwU wb‡j web¨vm 6C1 wU wb‡j web¨vm 6C2 NwU wb‡j web¨vm 6C3 OwU wb‡j web¨vm 6C4 cwU wb‡j web¨vm 6C5 5wU wb‡j web¨vm 6C6 Gfv‡e $+ wZwU †ei K‡i !-i 0‡jv †v> K‡i w.‡jB n‡q v‡e/ 0ikZu$3 % w"H3;q we#q4 KLb †v> Ki‡Z n‡e Avi KLb 03 Ki‡Z n‡e Lb $y ik# gwnjv wg‡j GKwUB .j n‡e ZLb Zv‡.i DjvDj .y wU 03 Ki‡Z n‡e/ wK&1 Lb .y wU wfb wfb .j _vK‡e ZLb †v> Ki‡Z n‡e/ $ieZ;% ‡Z mgq K‡i G we#‡q we¯Í vwiZ wjL‡ev/
∆
16
g‡@) 5 ] f1 O!"¨vqL mgv‡e%P
c~Y gvbL © 1f mgqL 1fwgwb 1E 3 Rb e¨wN 5_‡+ +Z7v‡e / R‡bi `) M:b +iv 5.‡Z cv‡iQ K L N >c 75 /E Rb 5)v+ cI ‡Z¨‡+ cI ‡Z¨‡+i mv‡_ +ig`© b +‡i- +ig`© b m48¨v +ZQ KC LN > c 7 3EG+w cvw© ‡Z w+9J 5)v+ Acw' Z w9)- Zviv cI ‡Z¨‡+ cI ‡Z¨‡+i mv‡_ ,¨v‡%+ +ivq ‡gv 66 w ,¨v‡%+ ,‡)v- n cvw© ‡Z 5gv +Z Rb Acw' Z w9)QQ GiKg !‡ovfv‡eF Avm‡Z $v‡iS KC LCc >C 7C XE 6w w7bW i‡i cZv+vi G+w ev G+vw"+w G+evi wb‡q +Zw m4‡+Z 5`qv .v‡eQ KCc LCcc5 >Cc5 7‡KvbwUB bq Encr= +ZQ KS
n! (n - r)! r!
LS
r (n - 1)!
>S (r-1)! ncr
7S
n! (n - r)!
AUigv)vL
g‡@) 5]f1 Omgv‡e%P 1E 8
/E+
3E +
XE M
E +
6.
7.
8.
9.
10.
tM) cvw8 $ !my ' ¨ +v+ GKevi GK †jvK $vnv(; GjvKvq †e(v‡Z †>j/ 7y i‡Z 7y i‡Z n[vZ †m GKwU KvK †.L‡jv vi .y wU @vbvB KvUv wQj/ KvKwUi G Ae¯8 v †.‡L †m f;#3 .y|L †$‡q g‡b g‡b fve‡jv, GUv wb}qB ‡Kv‡bv .y 2 †Q‡ji KvR/ †m g‡b g‡b ej‡Z jv>‡jv,~ nvq Avj vnM GB KvKwU GLb !(‡e wKfv‡e Avi w. †m Zvi LveviB m)>+ n Ki‡Z bv $v‡i Z‡e †m evW P‡e wKfv‡e~ ‡m Lb Gme fvewQ‡jv w[K ZLbB †mLv‡b GKwU •>j !‡( G‡jv vi †[W v‡U wKQY Lvevi wQ‡jv/ Lvevi0‡jv †m Kv‡Ki mvg‡b †Dj‡jv Ge) †mLvb †_‡K !‡( P‡j †>j/ GB .\ "¨ †.‡L †mB †jvKwU AZ¨&Í AevK n‡q †>j/ †m fve‡jv, ~ w. Gfv‡eB Avj vn ZvW i m\ w2‡K evW wP‡q iv‡Lb Z‡e Avgvi G‡Zv K2 K‡i KvR Kivi wK .iKvi Avwg AvR †_‡K †Kv‡bv KvR Ki‡ev bv, wZwbB Avgv‡K LvFqv‡eb/ ‡m KvR Kiv G‡Kev‡i e€ K‡i w.‡jv/ wK&1 .y B:wZb w.b $vi n‡q †>‡jF †m †Kv_vF ‡_‡K †Kv‡bv mvnv¨ †$j bv/ Gi Kvi3 Rvb‡Z †m GKRb vb; †jv‡Ki Kv‡Q †>j/ vb; †jvKwU mewKQY †"vbvi $i ej‡jb, ~ ZY wg †Zv .y wU $vwLB †.‡LwQ‡j/ GKwU †mB AvnZ KvK, Av‡iKUv †mB •>j/ wK&1 ZY wg †Kb †mB KvKwUi gZ n‡Z PvB‡j †Kb ZY wg †mB •>jwUi gZ n‡Z PvB‡j bv † wb‡Ri Lvevi‡Zv ‡v>v( K‡iB, mv‡_ viv bv †L‡q Av‡Q Zv‡.i gy ‡L Lvevi ZY ‡j †.q‚ $‡i Ae"¨ ‡jvKwU Zvi fY j ey '‡Z †$‡i •>j nFqvi $+ Z¨v"vq KvR Ki‡Z ik K‡iwQj/
♣ w"Hv4
∆
1
>‡ƒi d †jvKwUi gZB Avgv‡.i Pvi$v‡" A‡b‡KB Av‡Q viv A‡b¨i !$i wbf% i";j n‡q †eW ‡P _vK‡Z Pvq/ .y wU Kv‡Ri g‡a¨ mnR KvRwU Kivi R‡b¨ Zviv wb‡R‡.i g‡Zv K‡i GKwU ARy nvZ Zi; K‡i †bq/ Giv gvby ‡#i Rb¨ ‡Zv . ‡ii K_v, wb‡Ri Rb¨F wKQY Ki‡Z $v‡i bv/ Giv Ajm, Giv mgv‡Ri Rb¨ Kj¨v3Ki‡Zv bqB ei) †ev'v ¯= ik$/
m)>\ n;ZS
eB‡qi wVZ;q m)¯‡i _vK‡Q 4 $+ PY i B)‡iR; A)‡Ki m)‡vRb/ GB Aa¨vq .y wU †_‡K Avmv e¨v)‡Ki wewfb $+ ‡" i e¨vL¨vmn mgvavb / wewfb Kw[b $+ ‡" i mnR mij mgvavb GB Aa¨vq †_‡K wewmGm Gi wjwLZ $+ " Avm‡e ZvB wjwLZ $+ " F m)‡vRb Kivi †P2v Ki‡ev/ viv Avgvi eBwU m)>+ n Ki‡Z $vi‡eb bv Zviv Avgvi mv‡_ †Dmey ‡K †v>v‡v> Ki‡eb Avwg @KY ‡g„U w.‡q w.e/
ab¨ev. mevB‡K/
