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A standard six-sided die is rolled $6$ times. You are told that among the rolls, there was one $1,$ two $2$'s, and three $3$'s. How many possible sequences of rolls could there have been? (For example, $3,2,3,1,3,2$ is one possible sequence.)

 Feb 28, 2021
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This is the same as asking to permute the string \(333221\) in distinguishable ways. That would be equal to \(6!=6\cdot5\cdot4\cdot3\cdot2\), but we're overcounting by a factor of \(3! = 3\cdot2\) multiplied by \(2!=2\), so we need to divide that from the original 6!. That would be equal to \(\frac{6\cdot5\cdot4\cdot3\cdot2}{3\cdot2\cdot2} = \boxed{60}\)

 Feb 28, 2021

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