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(C) Let the marks assigned to the $8$ questions be $x_1, x_2, \dots, x_8$. We are given that $x_1 + x_2 + \dots + x_8 = 30$,where $x_i \ge 2$ for all

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$^{21}C_7$

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(C) Let the marks assigned to the $8$ questions be $x_1, x_2, \dots, x_8$. We are given that $x_1 + x_2 + \dots + x_8 = 30$,where $x_i \ge 2$ for all $i = 1, 2, \dots, 8$. Let $x_i = y_i + 2$,where $y_i \ge 0$. Substituting this into the equation,we get: $(y_1 + 2) + (y_2 + 2) + \dots + (y_8 + 2) = 30$ $y_1 + y_2 + \dots + y_8 + 16 = 30$ $y_1 + y_2 + \dots + y_8 = 14$. The number of non-negative integer solutions to this equation is given by the formula $^{n+r-1}C_{r-1}$,where $n=14$ and $r=8$. Number of ways $= ^{14+8-1}C_{8-1} = ^{21}C_7$.

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

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