Aptitude Questions and Answers

Permutations and Combinations

6. In how many ways can a group of 5 men and 2 women be made out of a total of 7 men and 3 women?
A. 1	B. 126
C. 63	D. 64

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Answer : Option C

Explanation :

We need to select 5 men from 7 men and 2 women from 3 women
Number of ways to do this
= ⁷C₅ x ³C₂
= ⁷C₂ x ³C₁ [Applied the formula ⁿC_r = ⁿC_{(n - r)} ]

=(7×62×1)×3

= 21 x 3 = 63

7. In how many different ways can the letters of the word 'MATHEMATICS' be arranged such that the vowels must always come together?
A. 9800	B. 100020
C. 120960	D. 140020

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Answer : Option C

Explanation :

The word 'MATHEMATICS' has 11 letters. It has the vowels 'A','E','A','I' in it and
these 4 vowels must always come together. Hence these 4 vowels can be grouped and
considered as a single letter. That is, MTHMTCS(AEAI).
Hence we can assume total letters as 8. But in these 8 letters, 'M' occurs 2 times,
'T' occurs 2 times but rest of the letters are different.

Hence,number of ways to arrange these letters = 8!(2!)(2!)=8×7×6×5×4×3×2×1(2×1)(2×1)=10080

In the 4 vowels (AEAI), 'A' occurs 2 times and rest of the vowels are different.

Number of ways to arrange these vowels among themselves = 4!2!=4×3×2×12×1=12

Hence, required number of ways = 10080 x 12 = 120960

8. There are 8 men and 10 women and you need to form a committee of 5 men and 6 women. In how many ways can the committee be formed?
A. 10420	B. 11
C. 11760	D. None of these

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Answer : Option C

Explanation :

We need to select 5 men from 8 men and 6 women from 10 women
Number of ways to do this
= ⁸C₅ x ¹⁰C₆
= ⁸C₃ x ¹⁰C₄ [Applied the formula ⁿC_r = ⁿC_{(n - r)} ]

=(8×7×63×2×1)(10×9×8×74×3×2×1)

= 56 x 210

= 11760

9. How many 3-letter words with or without meaning, can be formed out of the letters of the word, 'LOGARITHMS', if repetition of letters is not allowed?
A. 720	B. 420
C. None of these	D. 5040

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Answer : Option A

Explanation :

The word 'LOGARITHMS' has 10 different letters.
Hence, the number of 3-letter words(with or without meaning) formed by using these letters
= ¹⁰P₃
= 10 x 9 x 8
= 720

10. In how many different ways can the letters of the word 'LEADING' be arranged such that the vowels should always come together?
A. None of these	B. 720
C. 420	D. 122

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Answer : Option B

Explanation :

The word 'LEADING' has 7 letters. It has the vowels 'E','A','I' in it and
these 3 vowels should always come together. Hence these 3 vowels can be grouped
and considered as a single letter. that is, LDNG(EAI).
Hence we can assume total letters as 5 and all these letters are different.
Number of ways to arrange these letters = 5! = 5 x 4 x 3 x 2 x 1 = 120
In the 3 vowels (EAI), all the vowels are different.
Number of ways to arrange these vowels among themselves = 3! = 3 x 2 x 1= 6
Hence, required number of ways = 120 x 6= 720