The number of ways in which 21 identical apples can be distributed among three children such that each child gets at least 2 apples, is

This problem can be solved using the concept of combinations with repetition, also known as multiset coefficients or stars and bars.

Step 1: Since each child must get at least 2 apples, we can start by giving 2 apples to each child. This leaves us with 21 - 2*3 = 15 apples to distribute.

Step 2: The problem now reduces to finding the number of ways to distribute these 15 apples among the 3 children, which is a standard problem of distributing n identical items among r groups.

Step 3: The formula for this is C(n + r - 1, r - 1), where C is the combination function. In this case, n is 15 (the number of apples left) and r is 3 (the number of children).

Step 4: So, the number of ways to distribute the apples is C(15 + 3 - 1, 3 - 1) = C(17, 2).

Step 5: Calculate C(17, 2) = 17! / [2!(17-2)!] = 136.

So, there are 136 ways to distribute the 21 apples among the 3 children such that each child gets at least 2 apples.