Question

Find the number of ways of arranging all the letters of the word thunder so that the vowels appear in odd places

Answer 1

First arrange all letters except vowel in even places

And then in odd places arrange the vowels

Answer 2

Answer:

The number of ways of arranging all the letters of the word thunder so that the vowels appear in odd places is 1440.      

Step-by-step explanation:

To find : The number of ways of arranging all the letters of the word thunder so that the vowels appear in odd places?

Solution :

We have given word, THUNDER with 7 letters.

Vowels = 2 (U,E)

Consonants = 5 (T,H,N,D,R)

Lets name the positions of the  7  letters as  1,2,3,4,5,6,7.

There are 4 odd positions (1,3,5,7)

Select two odd positions as vowels are 2 from these  4  odd positions.

This can be done in $^4C_2$ ways.

Arrange the  2 vowels in the selected odd positions.  

This can be done in  2!  ways.

Arrange the  45 consonants in the remaining  5 positions.

This can be done in  5!  ways.

Total Number of ways is $^4C_2\times 2!\times 5!$

$=\frac{4!}{2!(4-2)!}\times 2\times 5\times 4\times 3\times 2$

$=\frac{4\times 3\times 2}{2\times2}\times 2\times 5\times 4\times 3\times 2$

$=3\times 2\times 2\times 5\times 4\times 3\times 2$

$=1440$

Therefore, The number of ways of arranging all the letters of the word thunder so that the vowels appear in odd places is 1440.