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So if I have five six sided dice what is the probability of getting three of a kind? Ie. 44452 or 63353.

I worked it out as

${{6 \choose 1}}$ ways of choosing the for the triple

${{5 \choose 3}}$ ways of choosing the dice for the triple

${{5 \choose 2}}$ ways of choosing the values for the single numbers

${{2 \choose 2}}$ ways of choosing the dice single numbers. (Only 1 one way of choosing 2 dice from remaining 2 dice).

$6^5$ total results from five six sided dice

So $$\frac{{{6 \choose 1}}{{5 \choose 3}}{{5 \choose 2}}{{2 \choose 2}}}{6^5}$$

And I get $\frac{25}{324}$...but it seems the actual answer is $\frac{25}{162}$ so where am I going wrong?

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1 Answer 1

When you count your "good" results, you count the outcomes $(4,4,4,2,5)$ and $(4,4,4,5,2)$ as a single result. But they are both counted when you say that the overall number of possible results is $6^5$. The point is, that you consider the dice being distinguishable. There fore the last point should read

  • $\binom 21$ ways to choose the die for the first single number, $\binom 11$ way to choose the die for the second single number

So you have the double number of "good" results, leaving you with an overall probability of $\frac {25}{162}$.

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