In how many ways can the letters of the word PERMUTATIONS be arranged if the
(i) words start with P and end with S,
(ii) vowels are all together,
(iii) there are always 4 letters between P and S?

Solution:

We know that the number of arrangements (permutations) that can be made out of n things out of which there are p, q, r, ... number of repetitions = n! / [p! q! r! ...].

In the given word PERMUTATIONS,

No. of T's = 2
Total number of letters = 12.

(i) It is given that words start with P and end with S. So the letters in the first and last positions are fixed. The middle 10 positions have to be filled with the remaining 10 letters among which there are 2 Ts (which are repeated).

No. of words = 10!/2! = 18,14,400.

Click here to see the formula behind it.

(ii) vowels are all together,

We know that there 5 vowels in the given letter. Then it becomes P, R, M, T, T, N, S, \(\fbox{EUAIO}\), so there are 8 units among which there are 2 Ts'.

Also, vowels can be interchanged within themselves in 5! ways.

Thus, the number of words  = 8!/2!× 5! = 24,19,200.

(iii) It is given that there are always 4 letters between P and S. So P and S can take the following positions respectively.

P	S
1st	6th
2nd	7th
3rd	8th
4th	9th
5th	10th
6th	11th
7th	12th

There are in total 7 ways in which there are 4 letters between P and S.

If we interchange P and S in the above table, we get 7 more ways of placing P and S.

Thus, total number of ways in which P and S can be placed = 7 + 7 = 14.

Now, the remaining 10 positions have to be filled with the remaining 10 letters among which there are 2 Ts. So

No. of ways for filling the remaining 10 positions = 10!/2!.

Total no. of ways = 14×10!/2! = 2,54,01,600

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NCERT Solutions Class 11 Maths Chapter 7 Exercise 7.3 Question 11

In how many ways can the letters of the word PERMUTATIONS be arranged if the (i) words start with P and end with S, (ii) vowels are all together, (iii) there are always 4 letters between P and S?

Summary:

(i) No. of ways = 18,14,400; (ii) If we consider all the 5 vowels as 1 unit, then no. of ways = 24,19,200; (iii) Total no. of ways = 2,54,01,600