! #$ %& '()*+ ,-.+(/*01 +.22*+.3.4 ),1*02 5*65 )+5003 )+*,1+,7 /,+51030687 ,16*1,,2*16 (1- 4(/5,4(/*+) 9)/,4: );,+*(3*<,- ).'=,+/ –
6?@ABC %% 6?@ABC %% +$?B )FGHBI# /J#KBC ;+(<=50/808> /J#KBC ;+(<=50/808>
)BDBE#B?C D*+># )BDBE#B?C D*+># '(@(>#(+ 1$L $M 5$F?EN )BDBE#B?C EA )BDBE#B?C EA .$8+>F >(@(>#(+ ;?BO?BPFJEJ#B 9JM QBBABA:C
)FGHBI# -BEI?JR#J$QC G# #.( ( )7 $C #.( /$8+>(H #.( >#87()#> @8># 9( 590( #$ 5440I /$)/(4#> 5)7 >$0J( 4+$90(@> *)J$0J*)- /$)*/ >(/#*$)>H >I>#(@> $C )$)0*)(5+ (K85#*$)>H >(+*(> 5)7 @5#.(@5#*/50 *)78/#*$)H /*+/805+ 5)7 #+*-$)$@(#+*/ C8)/#*$)>H #+*-$)$@(#+*/ *7()#*#*(>H 5)7 4$05+ /$$+7*)5#( >I>#(@L
+01/,1/ (Q@KT#JI 6B$DB#?T
+01/,1/ )/(1-(2-) 1.( 0(5+)(+> 7(@$)>#+5#( 5) 8)7(+>#5)7*)$CLLL
M(I /$)/(4#> $C /$)*/ >(/#*$)> 5)7 >I>#(@> $C )$)0*)(5+ (K85#*$)>
;,2S024(1+, )/(1-(2-) 1.( 0(5+)(+> >.500 9( 590( #$LLL
3,(21*16 +04;,/,1+*,) 1.( 0(5+)(+>LLL
%L @$7(0 >*#85#*$)> 544+$4+*5#(0I 5)7 >$0J( 4+$90(@> 5//8+5#(0I 8>*)/$)*/ >(/#*$)> 5)7 >I>#(@> $C )$)0*)(5+ (K85#*$)>
+0-,
)/,4U;+%%(6O*@O%
&L
*008>#+5#( #.( 7*CC(+()# #I4(> $C /$)*/ >(/#*$)>N 45+59$05H (00*4>(H /*+/0(H .I4(+9$05H 5)7 7(-()(+5#( /5>(>LOOO 7(C*)( 5 /*+/0(L
BL
7(#(+@*)( #.( >#5)75+7 C$+@ $C (K85#*$) $C 5 /*+/0(
)/,4U;+%%(6O*@OV
PL
-+54. 5 /*+/0( *) 5 +(/#5)-805+ /$$+7*)5#( >I>#(@
)/,4U;+%%(6O*@O>
7(C*)( 5 45+59$05 7(#(+@*)( #.( >#5)75+7 C$+@ $C (K85#*$) $C 5 45+59$05 -+54. 5 45+59$05 *) 5 +(/#5)-805+ /$$+7*)5#( >I>#(@ 7(C*)( 5) (00*4>( 7(#(+@*)( #.( >#5)75+7 C$+@ $C (K85#*$) $C 5) (00*4>( -+54. 5) (00*4>( *) 5 +(/#5)-805+ /$$+7*)5#( >I>#(@ 7(C*)( 5 .I4(+9$05 7(#(+@*)( #.( >#5)75+7 C$+@ $C (K85#*$) $C 5 .I4(+9$05
)/,4U;+%%(6O*@OW )/,4U;+%%(6O*GO% )/,4U;+%%(6O*GO& )/,4U;+%%(6O*IO% )/,4U;+%%(6O*IO& )/,4U;+%%(6O*IOV )/,4U;+%%(6O*AO% )/,4U;+%%(6O*AO&
QL RL SL EL TL %AL %%L %&L
! #$ %& '()*$+ ,*-. '/.$$0 '123 '4(/*50*6(7 '89:(/# – ;+(<=50/808> ?(/(@9(+ &A%B
)/,4U;+%%(6O*@O&
;5-( % $C $C
>
! #$ %& '()*+ ,-.+(/*01 +.22*+.3.4 ),1*02 5*65 )+5003 )+*,1+,7 /,+51030687 ,16*1,,2*16 (1- 4(/5,4(/*+) 9)/,4: );,+*(3*<,- ).'=,+/ –
+01/,1/
+01/,1/ )/(1-(2-)
;,2S024(1+, )/(1-(2-)
3,(21*16 +04;,/,1+*,) %BL
-+54. 5 .I4(+9$05 *) 5 +(/#5)-805+ /$$+7*)5#( >I>#(@
)/,4U;+%%(6O*AOV
%PL
+(/$-)*6( #.( (K85#*$) 5)7 *@4$+#5)# /.5+5/#(+*>#*/> $C #.( 7*CC(+()# #I4(> $C /$)*/ >(/#*$)>
)/,4U;+%%(6O*BO%
%QL
>$0J(> >*#85#*$)50 4+$90(@> *)J$0J*)- /$)*/ >(/#*$)>
)/,4U;+%%(6O*BO&
%RL
*008>#+5#( >I>#(@> $C )$)0*)(5+ (K85#*$)>
)/,4U;+%%(6O*MO%
%SL
7(#(+@*)( #.( >$08#*$)> $C >I>#(@> $C )$)0*)(5+ (K85#*$)> 8>*)#(/.)*K8(> >8/. 5> >89>#*#8#*$)H (0*@*)5#*$)H 5)7 -+54.*)-OOO >$0J( >*#85#*$)50 4+$90(@> *)J$0J*)- >I>#(@> $C )$)0*)(5+ (K85#*$)>
%EL )B?JBE @QA 4@#YBD@#JI@K *QAFI#J$Q
M(I /$)/(4#> $C >(+*(> 5)7 @5#.(@5#*/50 *)78/#*$) 5)7 #.( U*)$@*50 1.($+(@L
M(()0I $9>(+J( 5)7 *)J(>#*-5#( 45##(+)>H 5)7 C$+@805#( 544+$4+*5#( @5#.(@5#*/50 >#5#(@()#> 5)7 4+$J( #.(@ 8>*)@5#.(@5#*/50 *)78/#*$) 5)7F$+ U*)$@*50 1.($+(@L
+0-,
%L
*008>#+5#( 5 >(+*(>
&L BL PL QL
7*CC(+()#*5#( 5 >(+*(> C+$@ 5 >(K8()/( 8>( #.( >*-@5 )$#5#*$) #$ +(4+(>()# 5 >(+*(> *008>#+5#( #.( ;+*)/*40( $C 35#.(@5#*/50 V)78/#*$) 5440I @5#.(@5#*/50 *)78/#*$) *) 4+$J*)- *7()#*#*(>
RL
*008>#+5#( Pascal’s Triangle in the expansion of ! C$+ >@500 4$>*#*J( *)#(-+50 J508(> $C 4+$J( #.( U*)$@*50 1.($+(@ 7(#(+@*)( 5)I #(+@ $C ! H W.(+( *> 5 4$>*#*J( *)#(-(+H W*#.$8# (X45)7*)>$0J( 4+$90(@> 8>*)- @5#.(@5#*/50 *)78/#*$) 5)7 #.( U*)$@*50 1.($+(@
SL EL TL
! #$ %& '()*$+ ,*-. '/.$$0 '123 '4(/*50*6(7 '89:(/# – ;+(<=50/808> ?(/(@9(+ &A%B
)/,4U;+%%(6O*MOXO% )/,4U;+%%(6O*XO& )/,4U;+%%)4*O*YO% )/,4U;+%%)4*O*YO& )/,4U;+%%)4*O*YOV )/,4U;+%%)4*O*YO> )/,4U;+%%)4*O*YOJO% )/,4U;+%%)4*O*JO& )/,4U;+%%)4*O*JOV )/,4U;+%%)4*O*HO% )/,4U;+%%)4*O*HO&
;5-( & $C
>
! #$ %& '()*+ ,-.+(/*01 +.22*+.3.4 ),1*02 5*65 )+5003 )+*,1+,7 /,+51030687 ,16*1,,2*16 (1- 4(/5,4(/*+) 9)/,4: );,+*(3*<,- ).'=,+/ –
+01/,1/ /?JX$Q$DB#?T
+01/,1/ )/(1-(2-) M(I /$)/(4#> $C /*+/805+ C8)/#*$)>H #+*-$)$@(#+*/ *7()#*#*(>H *)J(+>( #+*-$)$@(#+*/ C8)/#*$)>H 5)7 #.( 4$05+ /$$+7*)5#( >I>#(@
;,2S024(1+, )/(1-(2-) %L C$+@805#( 5)7 >$0J( 5//8+5#(0I >*#85#*$)50 4+$90(@> *)J$0J*)/*+/805+ C8)/#*$)>
&L 5440I 544+$4+*5#( #+*-$)$@(#+*/ *7()#*#*(> *) >$0J*)- >*#85#*$)50 4+$90(@>
BL C$+@805#( 5)7 >$0J( 5//8+5#(0I >*#85#*$)50 4+$90(@> *)J$0J*)544+$4+*5#( #+*-$)$@(#+*/ C8)/#*$)> PL C$+@805#( 5)7 >$0J( 5//8+5#(0I >*#85#*$)50 4+$90(@> *)J$0J*)- #.( 4$05+ /$$+7*)5#( >I>#(@
3,(21*16 +04;,/,1+*,) %L
+0-,
&L
*008>#+5#( #.( 8)*# /*+/0( 5)7 #.( +(05#*$)>.*4 9(#W(() #.( 0*)(5+ 5)7 5)-805+ @(5>8+(> $C 5 /()#+50 5)-0( *) 5 8 )*# /*+/0( /$)J(+# 7(-+(( @(5>8+( #$ +57*5) @(5>8+( 5)7 J*/( J(+>5
)/,4U;+%%/O**@O% )/,4U;+%%/O**@O&
BL
*008>#+5#( 5)-0(> *) >#5)75+7 4$>*#*$) 5)7 /$#(+@*)50 5)-0(>
)/,4U;+%%/O**@OV
PL
*008>#+5#( #.( 7*CC(+()# /*+/805+ C8)/#*$)>
)/,4U;+%%/O**GO%
QL RL SL
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
)/,4U;+%%/O**GO& )/,4U;+%%/O**IO%
EL TL %AL %%L %&L %BL %PL %QL %RL %SL %EL %TL &AL &%L &&L
)/,4U;+%%/O**IOAO% )/,4U;+%%/O**AO& )/,4U;+%%/O**BO% )/,4U;+%%/O**BO& )/,4U;+%%/O**BOV )/,4U;+%%/O**MO% )/,4U;+%%/O**MO& )/,4U;+%%/O**MOXO% )/,4U;+%%/O**XO& )/,4U;+%%/O**YO% )/,4U;+%%/O**YO& )/,4U;+%%/O**YOJO% )/,4U;+%%/O**JO& )/,4U;+%%/O**HO% )/,4U;+%%/O**HO& )/,4U;+%%/O**HOV
OOO'8--(>#*$) C$+ V=1<().5)/(7 0(>>$) W.() 5J5*0590( 5)7 W.(+( 544+$4+*5#(
! #$ %& '()*$+ ,*-. '/.$$0 '123 '4(/*50*6(7 '89:(/# – ;+(<=50/808> ?(/(@9(+ &A%B
;5-( V $C
>
! #$ %& '()*+ ,-.+(/*01 +.22*+.3.4 ),1*02 5*65 )+5003 )+*,1+,7 /,+51030687 ,16*1,,2*16 (1- 4(/5,4(/*+) 9)/,4: );,+*(3*<,- ).'=,+/ –
+$AB '$$Z 3BXBQA )@DRKBC
3,6,1-
)/,4U;+%%(6O*@O%
-04(*1N +04;01,1/
)(4;3,
\(5+)*)- G+(5 5)7 '#+5)7F '89:(/# $+ '4(/*50*65#*$)
'/*()/(H 1(/.)$0$-IH 2)-*)((+*)- 5)7 35#.(@5#*/> ;+(<=50/808>
[+57( \(J(0
[+57( %%
?$@5*)F=$)#()#F =$@4$)()#F 1$4*/
G)50I#*/ [($@(#+I
+0-,
G)50I#*/ [($@(#+I
G[
SJ?E# ,Q#?T
.RRB?I@EB 3B##B?NE
)/,4U;+%%(6
'(+*(> 5)7 35#.(@5#*/50 V)78/#*$)
'3V
1
1+*-$)$@(#+I
O 2$D@Q 1FDB?@K O](+$ *C )$ >4(/*C*/ K85+#(+
^85+#(+
D*+># ^85+#(+
*
3$[B?I@EB 3B##B?NE O;8# 5 .I4.() Y #$ *)7*/5#( @$+( #.5) 5 >4(/*C*/ W((M
_((M
_((M $)(
@
O (?@GJI 1FDGB?
=$@4(#()/I
*008>#+5#( #.( 7*CC(+()# #I4(> $C /$)*/ >(/#*$)>N 45+59$05H (00*4>(H /*+/0(H .I4(+9$05H 5)7 7(-()(+5#( /5>(>
! #$ %& '()*$+ ,*-. '/.$$0 '123 '4(/*50*6(7 '89:(/# – ;+(<=50/808> ?(/(@9(+ &A%B
% ;5-( > $C
>