There is a group of 10 objects, 2 red, 3 blue and 5 green.
The objects are indistinguishable.
In how many ways can they be arranged on a line?
As there are 3 groups of objects I did that:
$ 10! / (2! * 3! * 5!) $
So my result is 25.200
Is that correct?
I am reading a lot of material around, but I am still unsure.
thanks
Best Answer
To explain where your formula comes from and why it works, consider breaking it up into steps:
Step 1: choose where the reds go. There are $\binom{10}{2}$ ways of arranging the reds and not reds (ignoring the fact that the not reds are of multiple colors for the moment).
Step 2: of the spaces labeled for use by not-reds, choose which of those spaces will be occupied by blues: There are $\binom{10-2}{3}$ number of ways to do this.
Step 3: of the spaces labeled for use by not-reds and not-blues, choose which are occupied by greens: There are $\binom{10-2-3}{5}$ number of ways.
Thus, there are $\binom{10}{2}\cdot\binom{8}{3}\cdot\binom{5}{5} = \frac{10!~~~8!~~~5!}{2!8!3!5!5!0!} = \frac{10!}{2!3!5!}$ number of ways to accomplish this.
(remember that $0!=1$ by definition)