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

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In how many different ways can the letters of word JUDGE be arranged so that the vowels always come together?

Total number of letters = 5 Number of vowels = 2. If we consider both vowel as a one letter then,

Required number = 4! 2! = 48.

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.

14. In how many different ways can the letters of the word 'JUDGE' be arranged such that the vowels always come together?
A. None of these	B. 32
C. 48	D. 64

Answer: Option C

Explanation:

The word 'JUDGE' has 5 letters. It has 2 vowels (UE) and these 2 vowels should always come together. Hence these 2 vowels can be grouped and considered as a single letter. That is, JDG(UE).Hence we can assume total letters as 4 and all these letters are different. Number of ways to arrange these letters$= 4!=4×3×2×1=24$In the 2 vowels (UE), all the vowels are different. Number of ways to arrange these vowels among themselves$=2!=2×1=2$

Total number of ways $=24×2=48$

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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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