Determine the number of different ways.

Question

A classic counting problem is to determine the number of different ways that the letters of

"dumbbell"can be arranged. Find that number.

Answer 1

Solution1. There are 2 letters that repeats, the rest are singles. 2b, 2l, 1d,1u,1m,1e = 8 letters.

You get the answers by 8!/(2!2!) = 10,080

Here you divide by 2! twice because of the repetitions.

Solution2.

There are 8 spots and you have to arrange your letters in them. First, you have C(8,2) ways to lay down your 2 b's, then you have 6 spots left. You have C(6,2) ways to place the 2l's - 4 spots remaining. Then you have 4! ways to arrange the 4 different letters.

C(8,2)*C(6,2)*4! = 10,080 different ways.