The number of ways in which an examiner can asign 30 marks to 8 questions giving not less than 2 marks in any question

^{20} C_{7}

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^{30} C_{8} -^{30} C_{2}

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^{21} C_{14}

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None of these

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Solution

The correct option is C $^{21} C_{14}$
The problem is same as the no. of positive integral solution of $q_{1} + q_{2} + q_{3} + . . . . . . . . + q_{8} = 30$ such that $q_{i} \geq 2 \forall i \in (1, 8)$
So let's distribute 2 to each qi , so now the equation becomes $q_{1} + q_{2} + q_{3} + . . . . . . . . + q_{8} = 30 - 16 \Rightarrow q_{1} + q_{2} + q_{3} + . . . . . . . . + q_{8} = 14$
Therefore, the no. of non-negative integral solution with constraint = $(\begin{matrix} 14 + 8 - 1 14 \end{matrix}) = (\begin{matrix} 2114 \end{matrix})$
Hence, option 'C' is correct.

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