First slide

Permutations

Question

How many different words can be formed by using all the letters o( the word 'ALLAHABAD' such that vowels occupy the even positions, is

Moderate

A

30
B

40
C

50
D

60

Solution

There are 4 vowels and all are alike i.e. 4A's. Also, there are 4 even places viz 2nd, 4th, 6th and 8th. So, these 4 even places can be occupied by 4 vowels in $\frac{4!}{4!} = 1$ way. Now, we are left with 5 places in which 5 letters, of which two are alike (2 L's) and other distinct, can be arranged in $\frac{5!}{2!}$ ways.
Hence, the total number of words in which vowels occupy the even places $= \frac{5!}{2!} \times \frac{4!}{4!} = \frac{5!}{2!} = 60$

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