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

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Exercise :: Permutation and Combination - General Questions

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13. 

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

A. 10080
B. 4989600
C. 120960
D. None of these

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.

How many different ways can the letters of the word psychology be arranged so that the vowels always come together?
Number of ways of arranging these letters =
8! = 10080.
(2!)(2!)

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

Number of ways of arranging these letters = 4! = 12.
2!

How many different ways can the letters of the word psychology be arranged so that the vowels always come together?
Required number of words = (10080 x 12) = 120960.


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Exercise :: Permutation and Combination - General Questions

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7. 

How many 3-digit numbers can be formed from the digits 2, 3, 5, 6, 7 and 9, which are divisible by 5 and none of the digits is repeated?

Answer: Option D

Explanation:

Since each desired number is divisible by 5, so we must have 5 at the unit place. So, there is 1 way of doing it.

The tens place can now be filled by any of the remaining 5 digits (2, 3, 6, 7, 9). So, there are 5 ways of filling the tens place.

The hundreds place can now be filled by any of the remaining 4 digits. So, there are 4 ways of filling it.

How many different ways can the letters of the word psychology be arranged so that the vowels always come together?
Required number of numbers = (1 x 5 x 4) = 20.

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