Question

The number of ways to distribute 30 identical candies among four children C1, C2, C3 and C4 so that C2 receives atleast 4 and atmost 7 candies, C3 receives atleast 2 and atmost 6 candies, is equal to:

• A. 205
• B. 615
• C. 510
• D. 430

Answer 1

Answer: (D)

430

Answer 2

The correct answer is (D) : 430
By multinomial theorem, no. of ways to distribute 30 identical candies among four children C1, C2 and C3, C4
= Coefficient of x30 in (x4 + x5 + … + x7) (x2 + x3 +…+ x6) (1 + x + x2…)2
=Coefficient of x24 in $\frac{(1−x^4)(1−x^5)(1−x^{31})^2}{(1−x)(1-x)(1-x)^2}$
= Coefficient of x24 in (1 – x4 – x5 + x9) (1 – x)–4
= 27C24 – 23C20 – 22C19 + 18C15 = 430