How many different ways can the letters of the word money be arranged so that the vowels always come together?

In how many different ways can the letters of the word 'JUDGE' be arranged in such a way that the vowels always come together?

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Answer & Explanation

Answer: Option C

Explanation:

In the word 'MATHEMATICS' we treat the vowels AEAI as one letter.

Thus, we have MTHMTCS (AEAI).

Now, we have to arrange 8 letters, out of which M occurs twice, T occurs twice and the rest are different.

Number of ways of arranging these letters = $$\frac{8 !}{(2 !) (2 !)}$$ = 10080.

Now, AEAI has 4 Letters in which A occurs 2 times and the rest are different.

Number of ways of arranging these letters = $$\frac{4 !}{2 !}$$ = 12.

$$\therefore$$ Required number of words = (10080 * 12) = 120960.

Solution:

The word ‘STRANGE’ has seven letters, including two vowels (A,E) and five consonants (S,T,R,N,G).

(i) the vowels come together?

If we consider two vowels to be one letter, we’ll end up with six letters that can be ordered in six different ways.

(A,E) can be combined in 2P2 ways.

As a result, the needed word count is

Using the formula, we can

$ P\text{ }\left( n,\text{ }r \right)\text{ }=\text{ }n!/\left( n-r \right)! $

$ P\text{ }\left( 6,\text{ }6 \right)\text{ }\times \text{ }P\text{ }\left( 2,\text{ }2 \right)\text{ }=\text{ }6!/\left( 6-6 \right)!\text{ }\times \text{ }2!/\left( 2-2 \right)! $

$ =\text{ }6!\text{ }\times \text{ }2! $

$ =\text{ }6\text{ }\times \text{ }5\text{ }\times \text{ }4\text{ }\times \text{ }3\text{ }\times \text{ }2\text{ }\times \text{ }1\text{ }\times \text{ }2\text{ }\times \text{ }1 $

$ =\text{ }720\text{ }\times \text{ }2 $

$ =\text{ }1440 $

(ii) the vowels never come together?

The total number of letters in the word ‘STRANGE’ is given by:

7P7 = 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1

7P7 = 5040

So,

Total number of words where vowels aren’t together = total number of words – total number of words in which vowels are always together

= 5040 – 1440 = 3600

As a result, there are 3600 configurations in which vowels never come together.