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Permutation And Combination - Aptitude MCQ Questions and Solutions with Explanations

Permutation And Combination MCQ Questions and answers with easy and logical explanations.Arithmetic Ability provides you all type of quantitative and competitive aptitude mcq questions on Permutation And Combination with easy and logical explanations. Permutation And Combination MCQ is important for exams like Banking exams,IBPS,SCC,CAT,XAT,MAT etc. Page-4 section-2

Each section contains 100 MCQs on Permutation and Combination .
31
In how many different ways can the letters of the word AWARE be arranged?
Discuss
Answer & Solution
Answer: Option B
Solution:
The given word contains 5 letters of which A is taken 2 times.
∴ Required number of ways
$$\eqalign{ & = \frac{{5!}}{{2!}} \cr & = \frac{{5 \times 4 \times 3 \times 2 \times 1}}{2} \cr & = 60 \cr} $$
32
A committee of 5 members is to be formed out of 3 trainees, 4 professors and 6 research associate. In how many different ways can this be done if the committee should have 2 trainees and 3 research associates?
Discuss
Answer & Solution
Answer: Option C
Solution:
Required number of ways
$$\eqalign{ & = \left( {{}^3{C_2} \times {}^6{C_3}} \right) \cr & = \left( {{}^3{C_1} \times {}^6{C_3}} \right) \cr & = \left( {3 \times \frac{{6 \times 5 \times 4}}{{3 \times 2 \times 1}}} \right) \cr & = 60 \cr} $$
33
In how many different ways can the letters of the word CREAM be arranged?
Discuss
Answer & Solution
Answer: Option B
Solution:
The given word contains 5 letters, all different.
∴ Required number of ways
$$\eqalign{ & = 5! \cr & = \left( {5 \times 4 \times 3 \times 2 \times 1} \right) \cr & = 120 \cr} $$
34
In how many different ways can the letters of the word ALLAHABAD be arranged?
Discuss
Answer & Solution
Answer: Option C
Solution:
The given word contains 9 letters, namely 4A, 2L, 1H, 1B and 1D.
∴ Required number of ways
$$\eqalign{ & = \frac{{9!}}{{4! \,2! \,1! \,1! \,1!}} \cr & = \frac{{9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}}{{4 \times 3 \times 2 \times 1 \times 2}} \cr & = 7560 \cr} $$
35
From a group of 7 men 6 women, 5 persons are to be selected to form a committee so that at least 3 men are there on the committee. In how many ways can it be done?
Discuss
Answer & Solution
Answer: Option D
Solution:
Required number of ways
$$ = \left( {{}^7{C_3} \times {}^6{C_2}} \right) + $$   $$\left( {{}^7{C_4} \times {}^6{C_1}} \right) + $$   $$\left( {{}^7{C_5} \times {}^6{C_0}} \right)$$
$$ = \left\{ {\frac{{7 \times 6 \times 5}}{{3!}} \times \frac{{6 \times 5}}{{2!}}} \right\}$$     $$ + \left( {{}^7{C_3} \times {}^6{C_1}} \right)$$   $$ + \left( {{}^7{C_2} \times 1} \right)$$
$$ = \left\{ {\frac{{7 \times 6 \times 5}}{6} \times \frac{{6 \times 5}}{{2 \times 1}}} \right\}$$     $$ + \left( {\frac{{7 \times 6 \times 5}}{{3 \times 2 \times 1}} \times 6} \right)$$    $$ + \left( {\frac{{7 \times 6}}{{2 \times 1}} \times 1} \right)$$
$$\eqalign{ & = \left( {525 + 210 + 21} \right) \cr & = 756 \cr} $$
36
In how many different ways can the letters of the word BANKING be arranged in such a way that the vowels always come together?
Discuss
Answer & Solution
Answer: Option E
Solution:
The given words contains 7 letters of which N is taken 2 times.
We keep the vowels (AI) together and treat them as 1 letter.
Thus, we have to arrange 6 letters BNKNG (AI) of which N occurs 2 times and rest are different.
These can be arranged in
$$\frac{6!}{2!}$$ = (6 × 5 × 4 × 3 ) = 360 days
Now 2 vowels (AI) can be arranged among themselves in 2 ways.
∴ Required number of ways = (360 × 2) = 720
37
A box contains 2 white, 3 black and 4 red balls. In how many ways can 3 balls be drawn from the box, if at least 1 black ball is to be included in the draw?
Discuss
Answer & Solution
Answer: Option C
Solution:
We may have (1 black and 2 non-black) or (2 black and 1 non-black) or (3 black).
Required number of ways
$$\eqalign{ & \left( {{}^3{C_1} \times {}^6{C_2}} \right) + \left( {{}^3{C_2} \times {}^6{C_1}} \right) + \left( {{}^3{C_3}} \right) \cr & = \left\{ {3 \times \frac{{6 \times 5}}{{2 \times 1}}} \right\} + \left( {\frac{{3 \times 2}}{{2 \times 1}} \times 6} \right) + 1 \cr & = \left( {45 + 18 + 1} \right) \cr & = 64 \cr} $$
38
Out of 5 men and 3 women, a committee of three members is to be formed so that it has 1 women, and 2 men. In how many different ways can it be done?
Discuss
Answer & Solution
Answer: Option D
Solution:
Required number of ways
$$\eqalign{ & = \left( {{}^3{C_1} \times {}^5{C_2}} \right) \cr & = 3 \times \frac{{5 \times 4}}{{2 \times 1}} \cr & = 30 \cr} $$
39
In how many different ways can the letters of the word BANANA be arranged?
Discuss
Answer & Solution
Answer: Option A
Solution:
The given words contains 6 letters of which A is taken 3 times, N is taken 2 times and the rest are all different.
∴ Required number of ways
$$\eqalign{ & = \frac{{6!}}{{3! \times 2!}} \cr & = \frac{{6 \times 5 \times 4 \times 3 \times 2 \times 1}}{{6 \times 2}} \cr & = 60 \cr} $$
40
A committee of 5 members is to be formed out of 3 trainees, 4 professors and 6 research associates. In how many different ways can this be done, if the committee should have 4 professors and 1 research associate or all 3 trainees and 2 professors?
Discuss
Answer & Solution
Answer: Option E
Solution:
Required number of ways
$$ = \left( {{}^4{C_4} \times {}^6{C_1}} \right) + $$   $$\left( {{}^3{C_3} \times {}^4{C_{2}}} \right)$$
$$\eqalign{ & = \left( {1 \times 6} \right) + \left( {1 \times \frac{{4 \times 3}}{2}} \right) \cr & = \left( {6 + 6} \right) \cr & = 12 \cr} $$