The number of ways in which an examiner can assign 30 marks to 8 questions, giving not less than 2 marks to…
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 :
${ }^{30} C_7$
${ }^{21} C_8$
${ }^{21} C_7$
${ }^{30} C_8$
Solution
30 marks to be alloted to 8 questions. Each question has to be given $\geq 2$ marks Let questions be $a, b, c, d, e, f, g, h$ and $a+b+c+d+e+f+g+h=30$
Let $a=a_1+2$ so, $a_1 \geq 0$ $b=a_2+2$ so, $a_2 \geq 0, \ldots \ldots . a_8 \geq 0$
So, $\left.\begin{array}{c}a_1+a_2+\ldots \ldots+a_8 \\ +2+2+\ldots . .+2\end{array}\right\}=30$ $\Rightarrow a_1+a_2+\ldots \ldots+a_8=30-16=14$
So, this is a problem of distributing 14 articles in 8 groups.
Number of ways $={ }^{14+8-1} \mathrm{C}_{8-1}={ }^{21} \mathrm{C}_7$