SUBMITTED SUBMIT TED BY: BY: APRIL GRACE L. CABULONG CABULONG
DAY 6
256.
In a class of 40 students, 27 like Calculus and 25 like Chemistry. How many like both Calculus and Chemistry? . !0 ". !! C. 12
#. !$ %olution& 'et ( ) number of students who like both sub*ects
27 + ( ( 25 + ( ) 40 27 5 + ( ) 40 ( ) !2
257.
club of 40 e(ecuti-es, $$ like to smoke arlboro and 20 like to smoke /hili orris. How many like both? . !0 ". !!
C. !2 D. 13
%olution& 'et ( ) number of e(ecuti-e who smoke both brand of ci1arettes
[33 – ( ( [20 – x] !0
$$ 3 ( ) 40 ( ) !$
25".
sur-ey of !00 ersons re-ealed re-ealed that 72 of them had eaten at restaurant restaurant / and that 52 of them had eaten at restaurant . hich of the followin1 could not be the number of ersons in the sur-eyed 1rou who had eaten at both / and ? . 20 ". 22 C. 2!
#. 26 %olution& 'et ( ) number of ersons who ha-e eaten in both restaurants restaurants
72 + ( ( 52 + ( ) !00
C. !2 D. 13
%olution& 'et ( ) number of e(ecuti-e who smoke both brand of ci1arettes
[33 – ( ( [20 – x] !0
$$ 3 ( ) 40 ( ) !$
25".
sur-ey of !00 ersons re-ealed re-ealed that 72 of them had eaten at restaurant restaurant / and that 52 of them had eaten at restaurant . hich of the followin1 could not be the number of ersons in the sur-eyed 1rou who had eaten at both / and ? . 20 ". 22 C. 2!
#. 26 %olution& 'et ( ) number of ersons who ha-e eaten in both restaurants restaurants
72 + ( ( 52 + ( ) !00
72 52 + ( ) !00 8 ) 24
25#.
9he robability robability of :C: :C: board e(aminees e(aminees from from a certain school school to ass the sub*ect athematics is $;7 and for the sub*ect Communications Communications is 5;7. if none of the e(aminees fails in both sub*ect and there are 4 e(aminees who ass both sub*ects,
%olution& 'et ( ) number of e(aminees who took the e(amination
3
8) 7 8
8)
7
x − 4 ¿
5
4
7
x − 4 ¿
x − 4
8 ) 2=
260.
In a commercial sur-ey in-ol-in1 !000 ersons in brand reference, !20 were found
to refer brand ( only, 200 refer brand y only, !50 refer brand > only, $70 refer either brand ( or y but not >, 450 refer brand y or > but not ( and $70 refer either brand > or ( but not y. How many ersons ha-e no brand reference, satis
". 2$0 C. !=0 #. !$0 %olution& 'et ( ) number of ersons who ha-e no brand reference
!000 ) ( !20 50 200 !00 !50 !00 8 ) 2=0
261.
toothaste
#. :ither yes or no %olution& ote& 9he sur-ey is not worth ayin1 for. Ane error is that accordin1 to the said sur-ey, there are 6 eole who used all three brands but only 5 eole used the brands Haee and Close3u.
262.
How many four3letter words be1innin1 and endin1 with a -owel without any letter reeated can be formed from the word BersonnelB? A. !0
". 4=0 C. 20 #. $!2 %olution& i-en word& /:D%A:' umber of -owels ) : and A umber of constants ) 5 /, D, %, and '
2
5
4
!
Eour letter word ote& ny of the two -owels can be
'et ) number of words ) 254! ) 40 ways
263.
Ei-e diFerent mathematics books, 4 diFerent electronics books and 2 diFerent communications books are to b laced in a shelf with the books of the same sub*ect to1ether. Eind the number of ways in which the books can laced. . 2G2 ". 5760 C. 3!560
#. !2=70 %olution& ath ath :lec :lec Comm. Comm.
:lec Comm. ath Comm. ath :lec
Comm. :lec Comm. ath :lec ath
%i( /atterns umber of ways the books in 9H can be arran1ed ) 5 umber of ways the books in :':C can be arran1ed ) 4 umber of ways the books in CA can be arran1ed ) 2 'et ) total number of ways ) 5 4 2 number of atterns ) 5 4 2 6 ) $4,560 ways
26!.
9he number of ways can $ nurses and 4 en1ineers can be seated on a bench with the nurses seated to1ether is .!44 ". 25= C. 720
#. 450 %olution& : : : :
: : :
: :
: :
: :
: : :
: : : :
Ei-e /atterns umber of ways the $ nurses can be arran1ed ) $ umber of ways the 4 en1ineers can be arran1ed ) 4 'et& n ) total number of ways n ) $ 4 number of atterns n ) $ 4 5 n ) 720 ways
265.
If 25 eole won ri>es in the state lottery assumin1 that there are no tiesJ, how many ways can these !5 eole win es? . 4,=45 ". !!6,260 C. 360%360
#. $,00$ %olution& !st ri> e !5
2nd ri> e !4
$rd ri> e !$
4th ri> e !2
5th ri> e !!
) !5 !4 !$ !2 !! ) $60,$60 ways
266.
How many 4 di1it numbers can be formed without reeatin1 any di1it from the followin1 di1its& !, 2, $, 4 and 6? A. 120
". !$0 C. !40 #. !50 %olution& !st di1it 5
2nd di1it 4
$rd di1it $
4th di1it 2
) 5(4($(2 ) !20 ways
267.
How many ermutations are there if the letters /DC%: are taken si( at a time? . !440 ". 4=0
C. 720
#. $60 %olution& ) n/n ) n ) 6 ) 720
26".
In how many ways can 6 distict books be arran1ed in a bookshelf? A. 720
". !20 C. $60 #. !=0 %olution& ) n/n ) n ) 6 ) 720 ways
26#.
hat is the number of ermutations of the letters in the word "? . $6 B. 60
C. 52 #. 42 %olution& i-en word& " umber of Ks ) $ umber of Ks ) 2
)
n! 6! = p ! q ! 3 ! 2 !
) 60 ways
270.
/%: unit has !0 :Ls, = /:Ls and 6 C/Ls. If a committee of $ members, one from each 1rou is to be formed, how many such committees can be formed? . 2,024 ". !2,!44 C. !"0
#. $60 %olution& ) !0
( 8) ( 6)
) 4=0 ways 271.
In how many ways can a /%: Chater with !5 directors choose a /resident, a Mice /resident, a %ecretary, a 9reasurer and an uditor, if no member can hold more than one osition? A. 360%360
". $2,760 C. $,00$ #. $,60$,600 %olution& /re s !5
M/
%ec
9rea s !2
!4
!$
)
15 ( 14 ) ( 13 ) ( 12 ) ( 11)
ud !!
) $60,$60
272.
Eour diFerent colored Na1s can be hun1 in a row to make coded si1nal. How many si1nals can be made if a si1nal consists of the dislay of one or more Na1s? A. 6!
". 66 C. 6= #. 62 %olution& ) 2/!
4/2 4/$ 4/4
4!
)
( 4 −1 ) !
4!
4!
( 4 −2 ) !
( 4 −3 ) !
4!
( 4 −4 ) !
) 64 si1nals
273.
In how many wayscan 4 boys and 4 1irls be seated alternately in a row of = seats? . !!52 ". 2$04 C. 576
#. 2204 %olution& umber of ways the 4 boys can be arran1ed ) 4 umber of ways the 4 1irls can be arran1ed ) 4 )
(4 !) (4 !)
) 576 ways
27!.
9here are four balls of four diFerent colors. 9wo balls are taken at a time and arran1ed in a de
#. $6 %olution& 4!
) 4/2 )
( 4 −2 ) !
) !2 ways
275.
How many diFerents ways can 5 boys and 5 1irls form a circle with boys and 1irls alternate? . 2=,=00 B. 2%""0
C. 5,600 #. !4,400 %olution&
umber of ways the boys can be arran1ed ) 5 3 !J ) 4 umber of ways the 1irls can be arran1ed ) 5 ) 4J5J ) 2==0 ways
276.
9here are four balls of diFerent colors. 9wo balls at a time are taken and arran1ed any way. How many such combinations are ossible? . $6 ". $ C. 6
#. !2 %olution& 4!
) 4C2 )
( 4 −2 ) ! 2 !
) 6 ways
277.
How many 63number combinations can be 1enerated from the numbers from ! to 42 inclusi-e, without reetition and with no re1ards to the order of the numbers?
. =50,66= B. 5%2!5%7"6
C. !==,=4=,2G6 #. $!,474,7!6 %olution& 42 !
)
42C6 )
( 42−6 ) ! 6 !
) 5,245,7=6 ways
27".
Eind the total number of combinations of three letters, O, D, 9 taken !, 2, $ at a time. A. 7
". = C. G #. !0 %olution& )
n
2
3!)
2
3
3 !
) 7 ways
27#.
In how many ways can you in-ite one or more of your <-e friends in a arty? . !5 B. 31
C. $6 #. 25 %olution&
n
2
)
3!)
2
5
3 !
) $! ways
2"0.
In how many ways can a committee of three consistin1 of two chemical en1ineers and one mechanical en1ineer can be formed from four chemical en1ineers and three mechanical en1ineers? A. 1"
". 64 C. $2 #. one of these %olution& umber of ways of selectin1 a chemical en1ineer&
N 1
4!
) 4C2 )
( 4 −2 ) ! 2 !
) 6
umber of ways of selectin1 a mechanical en1ineer&
N 2
3!
) $C! )
)6
(3)
( 3 −1 ) ! 1 !
) $
) != ways
2"1.
In athematics e(amination, a student may select 7 roblems from a set of !0 roblems. In how many ways can he make his choice? A. 120
". 5$0 C. 720
#. $20 %olution& 10 !
)
!0C7 )
( 10 −7 ) ! 7 !
) !20 ways
2"2.
How many committees can be formed by choosin1 4 men from an or1ani>ation of a membershi of !5 men? . !$G0 ". !240 C. !4$5 D. 1365
%olution& 15 !
)
!5C4 )
( 15 −4 ) ! 4 !
) !,$65 comminttees
2"3.
semiconductor comany will hire 7 men and 4 women. In how many ways can the comany choose from G men and 6 women who Puali
C. 4=0 #. =40 %olution& umber of ways of hirin1 men&
9!
N 1
) GC7 )
( 9 −7 ) ! 7 !
) $6
umber of ways of hirin1 women 6!
N 2
) 6C4 )
) $6
( 6− 4) ! 4 !
( 15 )=540
) !5
ways
2"!.
9here are !$ teams in a tournament. :ach team is to lay with each other only once. hat is the minimun number of dayss can they all lay without any team layin1 more than one 1ame in any day? . !! ". !2 C. 13
#. !4 %olution& 9otal number of 1ames 13 !
)
!$C2 )
( 13 −2 ) ! 2 !
) 7=
umber of 1ames that can be layed er day 13
)
2
) 6.5 ) 6 1ames;day
umber of days needed to comlete the tournament& 78
)
6
) !$ days
2"5.
9here are <-e main roads between the cities and ", and four between " and C. In how many ways can a erson dri-e from to C and return, 1oin1 throu1h " on both tris without dri-in1 on the same road twice? . 260 B. 2!0
C. !20 #. !60 %olution& umber of ways to tra-el from to " ) 5 umber of ways to tra-el from " to C ) 4 umber of ways to tra-el from C to " without usin1 the same road to tra-el from " to C ) 4 umber of ways to tra-el from " to without usin1 the same road to tra-el from to " ) 4
(4 )(3 )( 4)
)5
) 240 ways
2"6.
9here are 50 tickets in a lottery in which there is a e. hat is the robability of a man drawin1 a ri>e if he owns 5 tickets? . 50Q ". 25Q C. 20&
#. 40Q %olution& / ) robability of winnin1 a ri>e in the lottery 2
/)
50
1
)
25
/ ) robability for the man to win ) number of tickets he bou1ht ( robability of winnin1 a ri>e
( ) 1
/)5
25
/ ) 0.20
2"7.
Doll a air of dice. hat is the robability that the sum of two numbers is !!? . !;$6 ". !;G C. 1'1"
#. !;20 %olution& !
2
$
4
5
6
! 2 $ 4 5 6 umber of trials with a sum of !! ) 2 /)
number of successful trials total number of trials 2
/)
36
1
)
18
2"".
Doll two dice once. hat is the robability that the sum is 7?
A. 1'6
". !;= C. !;4 #. !;7 %olution& 9otal number of trials ) $6 !
2
$
4
5
6
! 2 $ 4 5 6 umber of trials with a sum of 7 ) 6
/)
number of successful trials total number of trials 6
/)
36
1
)
6
2"#.
In a throw of two dice, the robability of obtainin1 a total of !0 or !2 is . !;6 B. 1'#
C. !;!2 #. !;!= %olution& 9otal number of trials ) $6 umber of trials with a sum of !0 or !2 ) 4
!
2
$
4
5
6
! 2 $ 4 5 6
/)
number of successful trials total number of trials 4
/)
1
36
)
9
2#0.
#etermine the robability of drawin1 either a kin1 or a diamond in a sin1le draw from a ack of 52 layin1 cards. . 2;!$ ". $;!$ C. !'13
#. !;!$ %olution& ote& in a ack of 52 layin1 cards, there are 4 kin1 cards, !$ diamond cards and ! kin1 and diamond card at the same time.
P K
) robability of drawin1 a kin1
P D
) robability of drawin1 a diamond
P D ∧ K
) robability of drawin1 a kin1 at the same time diamond
P D ∨ K
)
P K + ¿
P D −¿
P D ∧ K
4
P D ∨ K
)
P D ∨ K
)
52
+
13 52
−
1 52
=
16 52
4 13
2#1.
card is drawn from a deck of 52 layin1 cards. Eind the robability of drawin1 a kin1 or a red card. . 0.5=$5 B. 0.53"5
C. 0.$5=5 #. 0.=5$5 %olution& ote& in a ack of 52 layin1 cards, there are 4 kin1 cards, 26 red cards and 2 kin1 and red card at the same time.
P K
) robability of drawin1 a kin1
P R
) robability of drawin1 a red card
P R
∧ K
) robability of drawin1 a kin1 at the same time a red card
P R
∨ K
)
P R
∨ K
P K + ¿ 4
)
P R ∨ K =¿
2#2.
52
+
26 52
0.5$=5
P R −¿
−
2 52
P R
∧ K
coin is tossed $ times. hat is the robability of 1ettin1 $ tails u? A. 1'"
". !;!6 C. !;4 #. 7;= %olution& 'et& / ) robability of 1ettin1 a head in a sin1le throw of a fair coin ) robability of 1ettin1 a tail in a sin1le throw of a fair coin r
/ ) nCr p q
n−r
1
here& )
R P )
2
( )( ) 1
/ ) $C$
1
2
3
1 2
0
)
2
(1)
R n ) $R r ) $
() 1
3
2
1
/)
8
2#3.
9he robability of 1ettin1 at least 2 heads when a coin is tossed four times is, . !!;!6 ". !$;!6 C. !;4 D. 3'"
%olution& r
/ ) nCr p q
n−r
1
here& )
1
R P )
2
( )( ) 1
p2 h
) 4C2
2
2
1
2
2
()
4!
p2 h
)
1
( 4 −2 ) ! 2 !
) 4C$
)
3
2
2
()
p4 h
)
1
2
2
() p¿
4
2
1
2
1
)
3 h +¿ p 4 h
/)
1
)
2
4
2
2 h +¿ p¿
4
( )( )
) 4C4
1
8
1
1
( 4 −3 ) ! 3 ! 1
p4 h
3
)
2
4!
p3 h
4
( )( ) 1
p3 h
R n ) 4R r ) 2
2
16
3
)
8
+
1 4
+
1 16
11
/)
16
2#!.
fair coin is tossed three times. hat is the robability of 1ettin1 either $ heads or $ tails? . !;= ". $;= C. 1'!
#. !;2 %olution& r
/ ) nCr p q
n−r
1
here& )
) $C$
/)
( )( ) 3
2
2
0
1 2
)
R n ) $R r ) $
( 1)
()
3
1 2
1
p3 h
p3 t
R P )
2
1
p3 h
p3 t
1
)
8
( )( ) 1
) $C$
3
2
1
0
2
)
( 1)
() 1
3
2
1
)
8
p3 h + p3 t
1
)
8
+
1 8
1
/)
4
2#5.
9he robability of 1ettin1 a credit in an e(amination is !;$. If three students are selected at random, what is the robability that at least one of them 1ot a credit? A. 1#'27
". =;27 C. 2;$ #. !;$ %olution&
ote& 1
/robability of 1ettin1 a credit )
3
2
/robability of not 1ettin1 any credit )
3
'et&
P1
) robability that only one student 1ot a credit
P1=¿
P2
$C!
1
3
2
2
3
3!
)
( )
12
4
( 3 −1 ) ! 1 !
27
)
27
) robability that e(actly two student 1ot a credit
P2=¿
P3
( )( ) 1
( )( ) 1
$C2
2
3
2
1
3
3!
)
( )
6
2
( 3 −2 ) ! 2 !
27
)
27
) robability that all three student 1ot a credit
P3=¿
$C$
( )( ) 1 3
3
2 3
( )
0
1
1
) !
27
)
27
/ ) robability that atleast one student 1ot a credit /)
P1+ P 2+ P3 =¿
12 27
+
6 27
+
1 27
19
/)
27
2#6.
9here are $ Puestions in a test. Eor each Puestion ! oint is awarded for a correct answer and none for a wron1 answer. If the robability that Oanine correctly answers a Puestion in the test is 2;$, determine the robability that she 1ets >ero in the test.
. =;27 ". 4;G C. !;$0 D. 1'27
%olution& ote& 9he only way that she can 1et a >ero is, if all her $ answer were wron1. 'et& ) robability of 1ettin1 a correct answer 2
)
3
P) robability of 1ettin1 a wron1 answer 1
P)
3
r
/ ) nCr p q
( )( ) 1
/ ) $C$
n−r
3
3
2 3
0
)
( 1)
() 1 3
3
1
)
27
2#7.
In the :C: "oard :(aminations, the robability that an e(aminee will ass each sub*ect is 0.=. hat is the robability that an e(aminee will ass at least 2 sub*ects out of the $ board sub*ects? . 70.GQ ". =0.GQ C. =5.GQ D. "#.6&
%olution& otes& /robability of 1ettin1 a assin1 score in each sub*ect is 0.= /robability of failin1 in any of the three sub*ects is 0.2 'et&
P1
) robability of assn1 e(actly two sub*ects
P1=¿
P1=
$C2
( 0.8 )2 ( 0.2 )1
3!
( 3−2 ) ! 2 !
( 0.128 )
) 0.$=4
P2
) robability of assin1 all three sub*ects
P2
) $C2
P2
) 0.5!2
( 0.8 )3 ( 0.2 )0
)
( 1 ) ( 0.512 )
/ ) robability of assin1 at least two sub*ects / ) 0.$=4 0.5!2 / ) 0.=G6
2#".
In a multily choice test, each Puestion is to be answered by selectin1 ! out of 5 choices, of which only ! is ri1ht. If there are !0 Puestions in a test, what is the robability o1 1ettin1 6 ri1ht of ure 1uesswork? . !0Q ". 6Q C. 0.44Q D. 0.55&
%olution& 'et& ) robability of 1ettin1 a correct answer 1
)
5
P ) robability of 1ettin1 a wron1 answer 4
P)
5
) robability of 1ettin1 6 correct answer out from !0 Puestions
( )( ) 1
/)
!0C6
5
6
4 5
10 !
/)
4
( 10 −6 ) ! 6 !
=
( )( ) 1 5
6
4
4
5
/ ) 0.0055
2##.
Erom a bo( containin1 6 red balls, = white balls and !0 blue balls, one ball is drawn at random. #etermine the robability that it is red or white. . !;$ B. 7'12
C. 5;!2 #. !;4 %olution& 'et& / ) robability of 1ettin1 a red or white ball from the bo(
number of ℜ d ∨¿ balls total numberof balls
/)
14
/)
=
24
7 12
300.
Erom a ba1 containin1 4 black balls and 5 white balls, two balls are drawn one at a time. Eind the robability that both balls are white. ssume that the
". !6;=! C. 5;!= #. 40;=! %olution& 'et&
P1=
5 9
P2=¿
robability of drawin1 a white ball in the second draw
ote& 9he !st ball was returned in the ba1 before the second ball was drawn
P2=
5 9
/ ) robability that both balls drawn are all white 5 5
/)
P1 x P2 = x 9
9
25
/)
81
301.
ba1 contains $ white and 5 black balls. If two balls are drawn in succession without relacement, what is the robability that both balls are black? . 5;!6 ". 5;2= C. 5;$2 D. 5'1!
%olution& 'et&
P1
) robability of drawin1 a black ball in the
P1=
P2
5 8
) robability of drawin1 a black ball in the second draw
ote& 9he !st ball was not returned in the ba1 before the 2 nd ball was drawn
P2=
4 7
/ ) robability that both ball drawn are all black 5 4
/)
P1 x P 2= x 8
20
/)
56
=
5 14
7
302.
n urn contains 4 black balls and 6 white balls. hat is the robability of 1ettin1 ! black and ! white ball in two consecuti-e draws from the urn? . 0.24 ". 0.27 C. 0.53
#. 0.04 %olution& ssume the
P1= P¿ x P¿
P1=
P1=
4 10
x
6 9
24 90
ssume the
P2= P¿ x P¿
P2=
P2=
4 10
x
6 9
24 90
'et& / ) robability that one ball is black and the other is white /)
P1+ P 2=
24 90
+
24 90
/ ) 0.5$$
303.
Erom a ba1 containin1 4 black balls and 5 white balls, two balls are drawn one at a time. Eind the robability that one ball is white and one ball is black. ssume that the
%olution& ssume the
P1= P¿ x P¿ 4
5
9
9
P1= x
P1=
40 81
ssume the
P2= P¿ x P¿ 4 5
P2= x 9
P2=
9
40 81
'et& / ) robability that one ball is black and the other is white
P1+ P 2=
/)
20 81
+
20 81
40
/)
81
30!.
1rou of $ eole enter a theater after the li1hts had dimmed. 9hey are shown to the correnct 1rou of seats by the usher. :ach erson holds a number stub. hat is the robability that each is in the correct seat accordin1 to the numbers in seat and stub? A. 1'6
". !;4 C. !;2 #. !;= %olution& 1
/robability that is correct )
3
, assumin1 he is to sit down
1
/robability that " is correct )
2
, assumin1 he is to sit down after
/robability that C is correct ) ! , assumin1 he is the last to sit down 'et& 1 1
/)
3
1
/)
305.
6
x x 1 2
Erom 20 tickets marked with
%olution& umbers from ! to 20, which is di-isible by $ ) 6 numbers
( 3,6,9,12,15,18)
umbers from ! to 20, which is di-isible by 7 ) 2 numbers
( 7,14 )
9otal numbers from ! to 20, which is di-isible by 7 or $ ) = numbers 'et& / ) robability that the ticket number is di-isible by $ or 7 /) 9otal numbers from ! to 20, which is di-isible by 7 or $ ) = numbers 'et& / ) robability that the ticket number is di-isible by $ or 7 /)
successful outcomes total outcomes 8
/)
20
2
/)
5
