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(A) Let $n_i$ be the marks assigned to the $i^{th}$ question,where $i = 1, 2, ..., 8$.We are given that $n_1 + n_2 + ... + n_8 = 30$ and $n_i \ge 2$ f

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$^{21}C_7$

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(A) Let $n_i$ be the marks assigned to the $i^{th}$ question,where $i = 1, 2, ..., 8$.We are given that $n_1 + n_2 + ... + n_8 = 30$ and $n_i \ge 2$ for all $i$.Let $x_i = n_i - 2$. Then $x_i \ge 0$.Substituting $n_i = x_i + 2$ into the sum equation:$(x_1 + 2) + (x_2 + 2) + ... + (x_8 + 2) = 30$$x_1 + x_2 + ... + x_8 + 16 = 30$$x_1 + x_2 + ... + x_8 = 14$,where $x_i \ge 0$.The number of non-negative integer solutions to this equation is given by the formula $\binom{n+r-1}{r-1}$,where $n = 14$ and $r = 8$.Number of ways = $\binom{14 + 8 - 1}{8 - 1} = \binom{21}{7}$.Thus,the correct option is $A$.

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

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