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Question: The number of ways in which an examiner can assign 30 marks to 8 questions, awarding not less than 2...

The number of ways in which an examiner can assign 30 marks to 8 questions, awarding not less than 2 marks to any question is

A

21C7

B

30C16

C

21C16

D

None of these

Answer

21C7

Explanation

Solution

Since the minimum marks to any question is two, the maximum marks that can be assigned to any questions is 16(=30 – 2 x 7), n1 + n2 + ....+ n8 = 30. If ni are the marks assigned to ith questions, then n1 + N2 + .....n8 = 30 with 2 ≤ ni ≤ 16 for i = 1,2,......,8. Thus the required number of ways

= the coefficient of x30 in (x2 + x3 + ....+ x16)8

= the coefficient of x30 in x16(1 + x + ...x14)8

= the coefficient of x30 in x16 (1−x151−x)8\left( \frac{1 - x^{15}}{1 - x} \right)^{8}

= the coefficient of x14 in (1 – x)-8. (1 – x15)8

= the coefficient of x14 in

{1+81!x+8.92!x2+8.9.103!x3+....}\left\{ 1 + \frac{8}{1!}x + \frac{8.9}{2!}x^{2} + \frac{8.9.10}{3!}x^{3} + .... \right\} (1 – 8C1 x15 + .....)

= the coefficient x14 in {1 + 8C1x + 9C2x2 + 10C3 x3 + ....}

Since the second bracket has powers of x0. x15 etc

= 21C14 = 21C7.