Question

The number of ways in which $30$ marks can be alloted to $8$ question if each carries at least $2$ marks is

• A. $115280$
• B. $117280$
• C. $116280$
• D. None of thes

Answer 1

Answer: (C)

$116280$

Required numbers

= coefficient of $x^{30}$ in ${(x^2+x^3+...x^{16})}^8$

= coefficient of $x^{30}$ in ${x^{16}(1+x......+x^{14})}^8$

=coefficient of $x^{14}$ in ${\left(\frac{1-x^{15}}{1-x}\right)}^8$

=coefficient of $x^{14}$ in ${(1-x)}^{-8}={{}^{21}C}_{14}=116280$