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Catherine rolls a standard 6-sided die six times. If the product of her rolls is 200, then how many different sequences of rolls could there have been? (The order of the rolls matters.)

 Apr 16, 2024
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Prime factorize 200: 200=23⋅52

To get a product of 200, Catherine must roll exactly 3 twos and 2 fives. There are 6 choices for the first die roll, then 5 choices left for the second, and so on. However, this overcounts the order she rolls the dice in (e.g., 2-5-2-5-2-2 is counted the same as 5-2-2-5-2-2).

To account for this, we divide by the number of times we've overcounted. Since every roll matters (there are no duplicate numbers besides the twos), we simply divide by the number of permutations of 6 items (where order matters). There are 6!=720 ways to order 6 distinct items.

So, the total number of winning sequences is:

7206⋅5⋅4⋅3⋅2⋅1​=5​

 Apr 16, 2024

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