Question

How many 5 - letter combinations can be made from the word STAIR : a . With no restrictions b . If the vowels cannot be side by side

Answer 1

The given word is STAIR.

Question (a)

We have to find how many 5 letter combinations can be made from the word STAIR.

Now, in the word, there are 5 distinct letters.

These 5 letters can be arranged among themselves in 5 places of the word, in 5! number of ways.

5!=5*4*3*2*1=120

From the word STAIR, 120 five-letter combinations can be made.

Question (b)

Now, the vowels cannot be side by side.

There are two vowels in the word, namely A and I.

Now, first let us find the number of ways in which these 2 vowels stay side by side.

Now, if we consider the two vowels as a single entity, there are 4 distinct entities; S,T,R and AI.

These 4 can permutate among themselves in 4! number of ways.

The two vowels can permutate amog themselves in 2 ways.

So, the total number of 5-letter combinations in which the vowels are together, is

4!*2, ie. 48.

Now, total number of 5-letter words is 120.

Total number of 5-letter words, in which the vowels are together, is 48.

So, the number of words in which the vowels are not together, is 120-48, ie. 72.

The answer is 72.