An examiner can assign 30 marks to 8 question | vedclass

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(A) Let $x_i$ be the marks assigned to the $i$-th question,where $i = 1, 2, \dots, 8$.We are given that $x_1 + x_2 + \dots + x_8 = 30$ and $x_i \ge 2$

Vedclass · Accepted Answer

$^{21}C_7$

Vedclass · Answer

(A) Let $x_i$ be the marks assigned to the $i$-th question,where $i = 1, 2, \dots, 8$.We are given that $x_1 + x_2 + \dots + x_8 = 30$ and $x_i \ge 2$ for all $i$.Let $y_i = x_i - 2$,so $y_i \ge 0$.Substituting $x_i = y_i + 2$ into the equation:$(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 is given by the formula $^{n+r-1}C_{r-1}$,where $n = 14$ and $r = 8$.Number of ways = $^{14+8-1}C_{8-1} = ^{21}C_7$.

An examiner can assign $30$ marks to $8$ questions. If he does not assign less than $2$ marks to any question,in how many ways can he assign the marks?

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