XAT Permutation and Combination Questions [Download PDF]
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Question 1: Letters of the word DIRECTOR are arranged in such a way that all the vowel come together .Find the No of ways making such arrangement?
a) 4320
b) 720
c) 2160
d) 120
e) None of these
1) Answer (A)
Solution:
Word – DIRECTOR
So “I,E,O” are there are 3! ways to arrange the vowels
Now “D,R,C,T,R” are the remaining alphabets ,
Condition is that the vowels should always be together so we can assume the vowels as a single alphabet/unit say “X” (‘X’=’I,E,O’) so now we have a new word – “D,R,C,T,R,X”
Possible arrangements for this word = 6!
Thus total number of ways to rearrange DIRECTOR with vowels grouped together = (Possible arrangements of ‘DRCTRX’) $\times$ (Possible arrangements of vowels)
= 6! $\times$ 3! = $720 \times 6 = 4320$
=> Ans – (A)
Question 2: In a box carrying one dozen of oranges one third have become bad.If 3 oranges taken out from the box random ,what is the probability that at least one orange out of the 3 oranges picked up is good ?
a) 1/55
b) 54/55
c) 45/55
d) 3/55
e) None of these
2) Answer (B)
Solution:
Total number of oranges in the box = 12
Number of ways of selecting 3 oranges out of 12 oranges, n(S) = $C^{12}_3$
= $\frac{12 \times 11 \times 10}{1 \times 2 \times 3} = 220$
Number of oranges which became bad = $\frac{12}{3}=4$
Number of ways of selecting 3 oranges out of 4 bad oranges = $C^4_3 = C^4_1 = 4$
Number of desired selection of oranges, n(E) = 220 – 4 = 216
$\therefore$ $P(E) = \frac{n(E)}{n(S)}$
= $\frac{216}{220}= \frac{54}{55}$
=> Ans – (B)
Instructions
Study the information carefully to answer the following questions:
A bucket contains 8 red, 3 blue and 5 green marbles.
Question 3: If 3 marbles are drawn at random, what is the probability that none is red ?
a) ${3 \over 8}$
b) ${1 \over {16}}$
c) ${1 \over {10}}$
d) ${3 \over {16}}$
e) None of these
3) Answer (C)
Solution:
Number of ways of drawing 3 marbles out of 16
$n(S) = C^{16}_3 = \frac{16 \times 15 \times 14}{1 \times 2 \times 3}$
= $560$
Out of the three drawn marbles, none is red, i.e., they will be either blue or green.
=> $n(E) = C^8_3 = \frac{8 \times 7 \times 6}{1 \times 2 \times 3}$
= $56$
$\therefore$ Required probability = $\frac{n(E)}{n(S)}$
= $\frac{56}{560} = \frac{1}{10}$
Question 4: If 2 marbles are drawn at random, what is the probability that both are green?
a) ${1 \over 8}$
b) ${5 \over {16}}$
c) ${2 \over 7}$
d) ${3 \over 8}$
e) None of these
4) Answer (E)
Solution:
Number of ways of drawing 2 marbles out of 16
$n(S) = C^{16}_2 = \frac{16 \times 15}{1 \times 2}$
= $120$
Out of the two drawn marbles, both are green
=> $n(E) = C^5_2 = \frac{5 \times 4}{1 \times 2}$
= $10$
$\therefore$ Required probability = $\frac{n(E)}{n(S)}$
= $\frac{10}{120} = \frac{1}{12}$
Question 5: If 4 marbles are drawn at random, what is the probability that 2 are red and 2 are blue ?
a) ${{11} \over {16}}$
b) ${3 \over {16}}$
c) ${11 \over {72}}$
d) ${3 \over {65}}$
e) None of these
5) Answer (D)
Solution:
Number of ways of drawing 4 marbles out of 16
=> $n(S) = C^{16}_4 = \frac{16 \times 15 \times 14 \times 13}{1 \times 2 \times 3 \times 4}$
= $1820$
Out of the four drawn marbles, 2 are red and 2 are blue.
=> $n(E) = C^8_2 \times C^3_2 = \frac{8 \times 7}{1 \times 2} \times \frac{3 \times 2}{1 \times 2}$
= $28 \times 3 = 84$
$\therefore$ Required probability = $\frac{n(E)}{n(S)}$
= $\frac{84}{1820} = \frac{3}{65}$
Instructions
From the following, differen committees are to be made as per the requirement given in each question.
In how many different ways can it de done? 10 men and 8 women out of which 5 men are teachers, 3 men doctors and businessmen. Among the women, 3 are 2 are teachers, 2 doctors, 2 researchers and 1 social worker.
Question 6: A Committee of 3 in which there is no teacher and no doctor.
a) 100
b) 120
c) 10
d) 12
e) None of these
6) Answer (C)
Solution:
Out of 10 men, Teachers = 5
Doctors = 3 and Business man = 2
Out of 8 women, Teachers = 3 and Doctors = 2
Researchers = 2 and Social worker = 1
Now, no teacher and doctor should be selected.
=> Remaining members (both men and women) = (2) + (2 + 1) = 5
Thus, number of ways of selecting 3 people out of 5
= $C^5_3 = \frac{5 \times 4 \times 3}{1 \times 2 \times 3} = 10$
Question 7: A Committee of 7.
a) 31824
b) 1200
c) 9600
d) 15912
e) None of these
7) Answer (A)
Solution:
There are 10 men and 8 women, Total = 18
Number of ways of selecting 7 members out of 18.
= $C^{18}_7 = \frac{18 \times 17 \times 16 \times 15 \times 14 \times 13 \times 12}{1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7}$
= $18 \times 17 \times 4 \times 13 \times 2 = 31824$
Question 8: A Committee is 5 in which 2 men teachers, 2 women teachers and 1 doctor are there.
a) 75
b) 150
c) 214
d) 20
e) None of these
8) Answer (B)
Solution:
Out of 10 men, Teachers = 5
Doctors = 3 and Business man = 2
Out of 8 women, Teachers = 3 and Doctors = 2
Researchers = 2 and Social worker = 1
Number of ways of selecting 2 men teachers, 2 women teachers and 1 doctor
= $C^5_2 \times C^3_2 \times C^5_1$
= $\frac{5 \times 4}{1 \times 2} \times \frac{3 \times 2}{1 \times 2} \times \frac{5}{1}$
= $10 \times 3 \times 5 = 150$
Question 9: A Committee of 4 in which at least 2 women are there.
a) 1260
b) 1820
c) 3060
d) 1890
e) None of these
9) Answer (D)
Solution:
Out of 10 men, Teachers = 5
Doctors = 3 and Business man = 2
Out of 8 women, Teachers = 3 and Doctors = 2
Researchers = 2 and Social worker = 1
Number of ways of selecting 4 members in which at least 2 are women
= (2 men , 2 women) + (1 men , 3 women) + (0 men , 4 women)
= $(C^{10}_2 \times C^8_2) + (C^{10}_1 \times C^8_3) + (C^{10}_0 \times C^8_4)$
= $(\frac{10 \times 9}{1 \times 2} \times \frac{8 \times 7}{1 \times 2}) + (\frac{10}{1} \times \frac{8 \times 7 \times 6}{1 \times 2 \times 3}) + (\frac{8 \times 7 \times 6 \times 5}{1 \times 2 \times 3 \times 4})$
= $(45 \times 28) + (10 \times 56) + (70)$
= $1260 + 560 + 70 = 1890$
Question 10: A Committee of 5 in which 3 men and 2 women are there.
a) 3360
b) 8568
c) 4284
d) 1680
e) None of these
10) Answer (A)
Solution:
Out of 10 men, Teachers = 5
Doctors = 3 and Business man = 2
Out of 8 women, Teachers = 3 and Doctors = 2
Researchers = 2 and Social worker = 1
Number of ways of selecting 3 men and 2 women.
= $C^{10}_3 \times C^8_2$
= $\frac{10 \times 9 \times 8}{1 \times 2 \times 3} \times \frac{8 \times 7}{1 \times 2}$
= $120 \times 28 = 3360$
