permutation and combination

11.

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?

Answer: Option A

Explanation:

Required number of ways = (⁷C₅ x ³C₂) = (⁷C₂ x ³C₁) =		7 x 6	x 3		= 63.
		2 x 1

12.

How many 4-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.	40	B.	400
C.	5040	D.	2520

Answer: Option C

Explanation:

‘LOGARITHMS’ contains 10 different letters.

Required number of words	= Number of arrangements of 10 letters, taking 4 at a time.
	= ¹⁰P₄
	= (10 x 9 x 8 x 7)
	= 5040.

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.

14.

In how many different ways can the letters of the word ‘OPTICAL’ be arranged so that the vowels always come together?

Answer: Option B

Explanation:

The word ‘OPTICAL’ contains 7 different letters.

When the vowels OIA are always together, they can be supposed to form one letter.

Then, we have to arrange the letters PTCL (OIA).

Now, 5 letters can be arranged in 5! = 120 ways.

The vowels (OIA) can be arranged among themselves in 3! = 6 ways.

Required number of ways = (120 x 6) = 720.