Permutation and Combination - General Questions

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

A.	120
B.	720
C.	4320
D.	2160
E.	None of these

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.

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.

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.

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.	63
B.	90
C.	126
D.	45
E.	135

Answer: Option A

Explanation:

Required number of ways = (⁷C₅ x ³C₂) = (⁷C₂ x ³C₁) = 1-sym-oparen-h1 7 x 6x 3 1-sym-cparen-h1 = 63.2 x 1

In how many different ways can the letters of the word 'DETAIL' be arranged in such a way that the vowels occupy only the odd positions?

A.	32
B.	48
C.	36
D.	60
E.	120

Answer: Option C

Explanation:

There are 6 letters in the given word, out of which there are 3 vowels and 3 consonants.

Let us mark these positions as under:

(1) (2) (3) (4) (5) (6)

Now, 3 vowels can be placed at any of the three places out 4, marked 1, 3, 5.

Number of ways of arranging the vowels = ³P₃ = 3! = 6.

Also, the 3 consonants can be arranged at the remaining 3 positions.

Number of ways of these arrangements = ³P₃ = 3! = 6.

Total number of ways = (6 x 6) = 36.

A box contains 2 white balls, 3 black balls and 4 red balls. In how many ways can 3 balls be drawn from the box, if at least one black ball is to be included in the draw?

A.	32
B.	48
C.	64
D.	96
E.	None of these

Answer: Option C

Explanation:

We may have(1 black and 2 non-black) or (2 black and 1 non-black) or (3 black).

Required number of ways

= (³C₁ x ⁶C₂) + (³C₂ x ⁶C₁) + (³C₃)

=		3 x	6 x 5		+		3 x 2	x 6		+ 1
			2 x 1				2 x 1

= (45 + 18 + 1)

= 64.

In how many ways a committee, consisting of 5 men and 6 women can be formed from 8 men and 10 women?

A.	266
B.	5040
C.	11760
D.	86400
E.	None of these

Answer: Option C

Explanation:

Required number of ways

= (⁸C₅ x ¹⁰C₆)

= (⁸C₃ x ¹⁰C₄)

=	8 x 7 x 6	x	10 x 9 x 8 x 7
	3 x 2 x 1		4 x 3 x 2 x 1

= 11760.

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?

A.	5
B.	10
C.	15
D.	20

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.

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