Ten prime digits with a set product

How many $10$ -digit numbers have all digits prime (each digit in $\{2, 3, 5, 7\}$ ) and a digit-product equal to exactly $100000$ ?

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$100000=2^5\cdot5^5$ . Digits are prime - $3$ s and $7$ s can't appear, so every digit is $2$ or $5$ .
Need exactly five $2$ s and five $5$ s in $10$ slots: $\binom{10}{5}$ .

Answer

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