How 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?

Exercise :: Permutation and Combination - General Questions

13.

In how many different ways can the letters of the word 'MATHEMATICS' be arranged so that the vowels always come together?

A.	10080
B.	4989600
C.	120960
D.	None of these

Answer: Option C

Explanation:

In the word 'MATHEMATICS', we treat the vowels AEAI as one letter.

Thus, we have MTHMTCS (AEAI).

Now, we have to arrange 8 letters, out of which M occurs twice, T occurs twice and the rest are different.

Number of ways of arranging these letters =	8!	= 10080.
(2!)(2!)

Now, AEAI has 4 letters in which A occurs 2 times and the rest are different.

Number of ways of arranging these letters =	4!	= 12.
2!

Required number of words = (10080 x 12) = 120960.

Page 2

Exercise :: Permutation and Combination - General Questions

How many 3-digit numbers can be formed from the digits 2, 3, 5, 6, 7 and 9, which are divisible by 5 and none of the digits is repeated?

Answer: Option D

Explanation:

Since each desired number is divisible by 5, so we must have 5 at the unit place. So, there is 1 way of doing it.

The tens place can now be filled by any of the remaining 5 digits (2, 3, 6, 7, 9). So, there are 5 ways of filling the tens place.

The hundreds place can now be filled by any of the remaining 4 digits. So, there are 4 ways of filling it.

Required number of numbers = (1 x 5 x 4) = 20.

Free

200 Qs. 200 Marks 120 Mins

Formula used:

If we have a group of 'n' objects out of which we make a selection taking 'r' object at a time, then the number of such selections is given by the combination formula

nCr = n!/(r!(n – r!))

Calculation:

Number of groups can be formed by 5 men and 2 women out of 7 men and 3 women

⇒ 7C5 × 3C2

⇒ 7!/(5! × (7 – 5)!) × 3!/(2! × (3 – 2)!)

⇒ 7!/(5! × 2!) × 3!/(2! × 1!)

⇒ (7 × 6 × 5!)/(5! × 2) × (3 × 2!)/(2!)

⇒ (7 × 3) × (3)

⇒ 63

∴ The number of ways to arrange men and women is 63

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Correct Answer:

Description for Correct answer:

There are 7 men and 3 women. We have to select 5 men out of 7 and 2 women out of 3. This can be done in \( \Large ^{7}C_{5} \times ^{3}C_{2} \) ways. The number of ways of making the selection = \( \Large ^{7}C_{5} \times ^{3}C_{2} \)= \( \Large ^{7}C_{2} \times ^{3}C_{2} \)\( \Large \left[ Because,\ ^{n}C_{r}=^{n}C_{n-r} \right] \)

= \( \Large \frac{7 \times 6}{1 \times 2} \times \frac{3 \times 2}{1 \times 2} \) = 63

Part of solved Permutation and combination questions and answers : >> Aptitude >> Permutation and combination

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