Expected Value of the Product of Frequencies for a Triangular Die Rolled 15 Times

We have a die with three sides (sides 1, 2, and 3). We roll this die 15 times and record the frequencies that each side shows up. What is the EV of the product of these frequencies?

For example, say we get $7$ ones, $3$ twos, and $5$ threes then the product is $7\cdot 3 \cdot 5=105.$


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Erdos Erdos
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