Question

How many words can be formed from the word "LUCKNOW" when the vowels always occupy even place
a.120
b.400
c.720
d.420​

Answer 1

420 ways

Step-by-step explanation:

given :

 LUCKNOW vowels should be in even places

Solution :

LUCKNOW it has 7 places now it has 3 even places and 2 vowels (U,O)

$==> 7! /3! * 2!\\==> 7*6*5*4*3*2*1 / 3*2*1*2*1\\==> 7*6*5*4/2*1\\==> 840 / 2\\==> 420 ways$

Answer 2

Given data: The word 'LUCKNOW'.

To Find: The number of words formed when the vowels occupy even place in the word 'LUCKNOW'.

Solution:

Number of letters in 'LUCKNOW' = 7 letters

Number of vowels (U,O) = 2 letters

Number of consonants (L,C,K,N,W) = 5 letters

Since, it is given the vowels occupy even place, the vowels can be placed in the positions of second, fourth and sixth.

Number of positions of vowels = 3

Thus, the permutation is done by,

$nP_{r}=\frac{n!}{(n-r)!}$

Substitute n =3, r =2

$3P_{2}=\frac{3!}{(3-2)!}$

$3P_{2} =3!$

$3P_{2}=3$×$2$×$1$

$3P_{2}=6$

The vowels can be arranged in 6 ways in even place.

The consonants can be placed in the positions of first, third, fifth, seventh.

Hence, the consonant can be placed in the positions of 5! ways

$5!=$$5$×$4$×$3$×$2$×$1$

$5!=120$

Thus, required number of ways = 6×120 = 720

Therefore, (c) 720 words can be formed from the word 'LUCKNOW' when the vowels always occupy even place.