How many ways can we distribute 43 bananas to 11 baby monkeys and 1 mother monkey, such that every baby monkey gets at least 2 bananas and the mother monkey gets at least 1?
I start by giving each baby monkey 2 bananas which leaves 21 bananas to be divided up. But will this be 21 choose 12 or 21 Choose 11 to calculate the answer?
Allocate the necessary $11 \cdot 2 + 1 = 23$ bananas first. Then there are $\binom{20+12-1}{12-1} = \binom{31}{11}$ ways to allocate the remaining $20$ bananas among the $12$ monkeys.