Combination and permutation of indistinguishable objects

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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

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1 Answer

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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)

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