Discrete math: how many ways are there to distribute 40 identical pieces of candy to 12 children so that every child is guaranteed to receive at least 2 candy?

13,037,895

Step-by-step explanation:

After you distribute 24 pieces of candy to the 12 children so each has 2 pieces, you have 16 remaining candies to distribute to 12 children. The "stars and bars" approach to counting the possibilities applies. In terms of binomial coefficients, you want to find ...

... C(16+12-1, 16) = 27!/(16!(27-16)!) = 13,037,895

On way to think about it is to consider that 11 dividers will divide something into 12 parts. In this case, the something is 16 candies. You essentially want to count the arrangements of 16 candies and 11 dividers. That number is 27 choose 16, or C(27, 16), as above.

(When dividers are adjacent, the child represented by the space between them gets 0 additional candies. One possible arrangement is that a given child gets all 16 remaining candies, and the other children get no additional candies.)