Find the number of permutations of letter of the word KAPIL
(i) When P is always in the middle.
(ii) When vowels occupy odd places
(i) When P is always in the middle:
To find the number of permutations when P is in the middle, we need to consider the remaining 4 letters (K, A, I, L) and their arrangements.
Now let's calculate the number of arrangements of these 4 letters by treating them as distinct objects:
Number of arrangements = 4! = 4 x 3 x 2 x 1 = 24
Therefore, when P is always in the middle, there are 24 possible permutations.
(ii) When vowels occupy odd places:
In the given word KAPIL, the vowels are A and I.
To find the number of permutations when the vowels occupy odd places, we need to consider the remaining 3 consonants (K, P, L) and their arrangements.
Now let's calculate the number of arrangements of these 3 consonants by treating them as distinct objects:
Number of arrangements = 3! = 3 x 2 x 1 = 6
Therefore, when the vowels occupy odd places, there are 6 possible permutations.
(i) When P is always in the middle:
To find the number of permutations when "P" is in the middle, we can treat "KAPIL" as one entity. Then, there are 5 entities in total: K, A, I, L, and the whole entity "KAPIL" with "P" in the middle.
The number of permutations of these 5 entities is equal to 5!. However, we also need to consider the permutations within the entity "KAPIL" itself.
Since P is fixed in the middle, we only need to arrange the remaining 4 entities (K, A, I, and L). Therefore, the number of permutations when P is in the middle is 4!.
So, the number of permutations when P is always in the middle is 4!.
(ii) When vowels occupy odd places:
The word KAPIL has 2 vowels (A and I) and 3 consonants (K, P, and L).
To place the vowels (A and I) at odd places (1st, 3rd, or 5th position), we need to consider the following cases:
1. A is at the 1st position and I is at the 3rd position:
In this case, we have the following arrangement: A _ I _ _
The consonants (K, P, and L) can be arranged in the remaining three positions in 3! ways.
2. I is at the 1st position and A is at the 3rd position:
In this case, we have the following arrangement: I _ A _ _
Again, the consonants (K, P, and L) can be arranged in the remaining three positions in 3! ways.
Therefore, the total number of arrangements when vowels occupy odd places is 2 * 3! = 2 * 6 = 12.
So, the number of permutations when vowels occupy odd places is 12.