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. ${}^{21}{C}_7$
• B. ${}^{20}{C}_7$
• C. ${}^{21}{C}_8$
• D. ${}^{20}{C}_8$

Answer 1

Answer: (A)

Let $x_i$ denote the marks assigned to the question.
Then $x_1+x_2+x_3+x_4+x_5+x_6+x_7+x_8=$
$30$ where $x_i\ge 2$, $i=1,2,3,4,.......8$
Let $y_i=x_i-2,i=1,2,,......8$
$\therefore y_1+y_2+......+y_8$
$=\left(x_1+x_2+......+x_g\right)-8\times 2=30-16=14$
Hence the total number of solutions of this equation is ${}^{14+8-1}{C}_{8-1}={}^{21}{C}_7$