Suppose that $10$ dice are rolled. Each die is a regular $6$ -sided die with numbers $1$ through $6$ labelled on the sides. How many different distinct sums of all 10 numbers are possible?

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Details and assumptions
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The sums $5 = 1+4$ and $5 = 2 + 3$ are the same sum (namely 5), and should only be counted once.

The answer is 51.

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