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) Well, let's put the letter "P" in the middle and see what mischief we can create. Now, we have 4 different letters left: K, A, I, and L. We can arrange these letters in 4! = 24 ways. So, there are 24 permutations when the letter P is always in the middle.
(ii) Ah, the vowels want to occupy odd places, huh? Well, let's humor them for a moment. We have 2 vowels: A and I. They need to be placed in the 1st, 3rd, or 5th position. We can arrange them in 3! = 6 ways. As for the remaining consonants, K, P, and L, they can be arranged in 3! = 6 ways as well. So, in total, we have 6 x 6 = 36 permutations when vowels occupy odd places.
Hope my clownish explanations brightened your day a bit!
(i) When P is always in the middle:
To find the number of permutations when the letter P is always in the middle, we need to fix the position of P and arrange the remaining letters.
Step 1: Fix the position of P in the middle.
Step 2: Arrange the remaining letters: K, A, I, and L.
Step 1: There is only one position available for P, which is the middle position.
Step 2: The remaining letters K, A, I, and L can be arranged in 4! = 4 x 3 x 2 x 1 = 24 ways.
So, the number of permutations when P is always in the middle is 1 x 24 = 24.
(ii) When vowels occupy odd places:
To find the number of permutations when the vowels occupy odd places, we need to arrange the consonants (K, P, L) and vowels (A, I) separately.
Step 1: Arrange the consonants (K, P, L).
Step 2: Arrange the vowels (A, I).
Step 1: The consonants (K, P, L) can be arranged in 3! = 3 x 2 x 1 = 6 ways.
Step 2: The vowels (A, I) can be arranged in 2! = 2 x 1 = 2 ways.
So, the number of permutations when vowels occupy odd places is 6 x 2 = 12.
To find the number of permutations of the letters in the word "KAPIL," we can use the concept of permutations with restrictions. Let's break down each case:
(i) When P is always in the middle:
In this case, we have three fixed letters: K, A, and L. We can arrange these three letters in 3! = 3 x 2 x 1 = 6 ways. P will go in the middle, and I can be placed in the first or last position. Hence, there are 2 places for I. Therefore, the number of permutations when P is always in the middle is 6 x 2 = 12.
(ii) When vowels occupy odd places:
In this case, we need to place the vowels (A and I) in the odd positions. We can consider the three odd positions (1st, 3rd, and 5th) as one group and the two even positions (2nd and 4th) as another group. Within each group, the letters can be arranged independently.
For the odd positions, we have two vowels, so they can be arranged in 2! = 2 x 1 = 2 ways.
For the even positions, we have two consonants (K and P), and they can be arranged in 2! = 2 x 1 = 2 ways.
So, the total number of permutations when vowels occupy odd places is 2 x 2 = 4.
Therefore, the number of permutations of the letters in the word "KAPIL" in each case is:
(i) When P is always in the middle: 12
(ii) When vowels occupy odd places: 4