The number of ways to distribute 30 identical candies among four children C_1,C_2,C_3,C_4 …

Question

The number of ways to distribute 30 identical candies among four children $C_1,C_2,C_3,C_4$ so that $C_2$ receives at least 4 and at most 7 candies, $C_3$ receives at least 2 and at most 6 candies, is equal to:

Accepted Answer

Using generating functions: coefficient of $x^{30}$ in $(x^4+\cdots+x^7)(x^2+\cdots+x^6)(1+x+x^2+\cdots)^2$

$$= [x^{24}]\ (1-x^4)(1-x^5)(1-x)^{-4} \cdot (1-x^{31})^2\text{ terms negligible}$$
$$= [x^{24}]\ (1-x^4-x^5+x^9)(1-x)^{-4}$$
$$= \binom{27}{24}-\binom{23}{20}-\binom{22}{19}+\binom{18}{15} = 2925-1771-1540+816 = 430$$

Solutions · 1