Five ordinary dies are rolled at random and the sum of numbers shown is $16$. What is the probability that the number shown on each is any one from $2,3,4$ or $5?$

Using generating functions, the number of ways to obtain a sum of 16 from five dice is the coefficient of x^16 in (x^2 + x^3 + x^4 + x^5)^5. This equals the coefficient of x^6 in (1 + x + x^2 + x^3)^5, which evaluates to 65. The total number of ways to obtain a sum of 16 from five unconstrained dice is the coefficient of x^11 in (1 + x + x^2 + x^3 + x^4 + x^5)^5, which evaluates to 735. The required probability is the ratio of favorable outcomes to total outcomes, 65/735, which simplifies by dividing the numerator and denominator by 15 to get 9/49.

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