The number of ways in which an examiner can a | vedclass

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Question

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

Vedclass · Accepted Answer

$^{21}C_7$

Vedclass · Answer

(C) Let the marks assigned to the $8$ questions be $x_1, x_2, \ldots, x_8$.
We are given that:
$x_1 + x_2 + \cdots + x_8 = 30$, where $x_i \ge 2$ for each $i \in \{1,2,\ldots,8\}$.
Let $x_i = y_i + 2$, where $y_i \ge 0$.
Substituting:
$(y_1 + 2) + (y_2 + 2) + \cdots + (y_8 + 2) = 30$
$y_1 + y_2 + \cdots + y_8 + 16 = 30$
$y_1 + y_2 + \cdots + y_8 = 14$
The number of non-negative integer solutions is given by:
$\binom{n+r-1}{r-1}$, where $n=14$ and $r=8$.
Number of ways:
$\binom{14+8-1}{8-1} = \binom{21}{7} = {}^{21}C_7$

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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