Permutation and Combination

Question:
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

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.

Question: In how many ways can the letters of the word ‘LEADER’ be arranged?
[A].

72

[B].

144

[C].

360

[D].

720

Answer: Option C

Explanation:

The word ‘LEADER’ contains 6 letters, namely 1L, 2E, 1A, 1D and 1R.

Required number of ways =	6!	= 360.
	(1!)(2!)(1!)(1!)(1!)

Video Explanation: https://youtu.be/2_2QukHfkYA

Question: 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.