Question: Pawan was designing a Mock TEST. He had 8 questions for this Mock test and he had to assign a total of 30 marks to the 8 questions in this Mock test. If the minimum marks are assigned to a question is 2 and each question carries integral marks, then how many ways are possible for Pawan to distribute 30 marks in this Mock test?

a. C(21,7)
b. C(23,7)
C. C(23,5)
d. C(11,5)
e. None of these

Solution:

We know that the number of non - negative integer solutions of the equation:

x1 + x2 + ... + xn = r is C(n + r - 1, r)

From the given problem, we can form the following equation:

x1 + x2 + ... + x8 = 30
Since the minimum marks assigned to each question is 2, we have:

x1 – 2 + x2 – 2 + ... + x8 – 2 = 30 – 16

or, x1 + x2 + ... + x8 = 14

So, now each of x1, x2, …, x8 are non negative integers

Hence, the number of ways is:
C(21, 14) and that is equal to C(21, 7) as we know that C(n, r) = C(n, n – r).

Thank You

Ravi Raja