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

Question

The number of ways to distribute 30 identical candies among four children C,, C_2, C_3 and 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

Correct answer: D. (d) t_2+t_2+t_3+t_4=30 Coefficient of x^30 in (1+x+x^2+ …+x^30)^2 (x^4+x^5+x^6+x^7)(x^2+x^3+x^4+x^5+x^6) x^6 ( 1-x^311-x )^2(1+x+x^2+x^3)(1+x+x^2+x^3+x^4) x^6(1-x^31)^2(1-x^4)(1-x^5) (1-x )^4 x^6(1-x^4-x^5+x^9)(1+x^62-2x^31 (1-x )^-4) x^6(1-x^4-x^5+x^9) (1-x )^-4 Coefficient of x_n^n in (1-x )^-r is ^n+r-1C_r-1 ⇒ ^27C_3- ^23C_3- ^22C_3+ ^18C_3 2925-1771-1540+816 =430 OR x_2 ∈ [4 ,7 ] ,x_3 ∈ [2 ,6 ] ⇒t_1+t_2+t_3+t_4=24 total ways ^24+4-1C_4-1- ^20+4-1C_4-1- ^19+4-1C_4-1+ ^15+4-1C_4-1 = ^27C_3- ^23C_3- ^22C_3+ ^18C_3=430

$The number of ways to distribute 30 identical candies among four children C,, C_{2}, C_{3} and C_{4} so that C_{2} receives at least 4 and at most 7 candies, C_{3} receives at least 2 and at most 6 candies, is equal to$

Solution

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