Question

how many ways can the letter of the word intermediate be arranged so that the vowels occupy even places

Answer 1

Given word is ' INTERMEDIATE '
the number of vowels in the word are 6 [ vowels are I , E , E, I, A , E ]
here we can see that , 3 times using E , 2 times using I and one time using A

Also, question said that we have to arrange vowels in even places
There are six even places in the word.
So, we have to arrange whole vowels in six places .
First of all vowels are arranged itself .
e.g., number of possible arrangement of vowels itself = 6!/3!×2! [ Because 3tines of E and 2 time of I appear in word ]
= 60
Now, 6 vowels arrange in six places = ⁶C₆ = 1

Now, rest six consonants are N, T, R, M, D , T [ T comes 2 times
So, number of possible arrangement of consonants itself = 6!/2! = 360
Number of ways of 6 consonants arranged in six places = ⁶C₆ = 1

Now, total number of arrangement = 1 × 60 × 360 × 1 = 21600 ways

Answer 2

Solution :-

There are 12 letters in the word INTERMEDIATE.

Out of 12 places the even places are 2nd, 3rd, 6th, 8th, 10th and 12th. There are 6 vowels in the given word. In the given word there are two I, three E and one A.

So, 

Number of ways to occupy the 6 even places with vowels = (6!)/(3!2!)

⇒ (6*5*4*3*2*1)/(3*2*1*2*1)

⇒ 720/12

= 60 ways

Number of ways to occupy remaining places = (6!)/(2!)

⇒ (6*5*4*3*2*1)/(2*1)

⇒ 720/2

= 360 ways

Total number of ways = 60*360

= 21600 ways

Answer.