Question

Can anyone tell me this question! this is related to permutations chapter.

Please help !!!

Question is -:
Write down all the permutations of the vowels A, E, I, O, U taking 3 at a time and starting
with A.

Answer 1

Here's my method. There are 5! ways to order the A,E,I,O,U. Next we need to decide in how many ways can we put the X's in between those vowels. We need at least three in between any of them, so that's already 4⋅3=12 X's, and hence we've got 3 X's left to distribute. We now have three options: (1) put all three X's between the same two vowels. This gives 4 possibilities. (2) Put two X's between the same pair of vowels and the other one between a different pair. This gives 4⋅3=12 possibilities. And finally (3) we could put all three X's between three different pairs of vowels. That gives

(43)=4!3!1!=4

possibilities as this is the number of ways we can choose 3 out of the 4 pairs. In total then, we get

5!⋅(4+12+4)=120⋅20=2400

possibilities, which agrees with CY Aries's answer.