# In how many different ways can the letters of the word 'GEOGRAPHY' be arranged such that the vowels must always come together?

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In how many different ways can the letters of the word 'GEOGRAPHY' be arranged such that the vowels must always come together?
1. 2520
2. 2530
3. 15130
4. 15120

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Correct Answer - Option 4 : 15120

Given:

The given number is 'GEOGRAPHY'

Calculation:

The word 'GEOGRAPHY' has 9 letters. It has the vowels E, O, A in it, and these 3 vowels must always come together. Hence these 3 vowels can be grouped and considered as a single letter. That is, GGRPHY(EOA).

Let 7 letters in this word but in these 7 letters, 'G' occurs 2 times, but the rest of the letters are different.

Now,

The number of ways to arrange these letters = 7!/2!

⇒ 7 × 6 × 5 × 4 × 3 = 2520

In the 3 vowels(EOA), all vowels are different

The number of ways to arrange these vowels = 3!

⇒ 3 × 2 × 1 = 6

Now,

The required number of ways = 2520 × 6

⇒ 15120

∴ The required number of ways is 15120.