Question

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:

• A. 11
• B. ${}^{21}{C}_7$
• C. 18
• D. 235

Answer 1

The correct option is B ${}^{21}{C}_7$

Let $x_i$ denote the marks assigned to the $i^{th}$ question. Then
$x_1+x_2+x_3+\cdots +x_7+x_8=30$
where i = 1, 2, 3, . . ., 8 and $x_i\ge 2$
and $y_1+y_2+\cdots +y_7+y_8=14$
where $y_i=x_i-2,i$ = 1, 2, 3 . . . .
$\therefore$ The total number of solutions of this equation is
${}^{14+8-1}{C}_{8-1}{=}^{21}{C}_7$