• # question_answer The number of ways in which an examiner can assign 30 marks to 8 question, giving not less than 2 marks to any question, is:     JEE Main  Online Paper (Held On 22 April 2013) A)  $^{30}{{C}_{7}}$                                B)  $^{21}{{C}_{8}}$ C)  $^{21}{{C}_{7}}$                                                D)  $^{30}{{C}_{8}}$

30 marks to be alloted to 8 questions. Each question has to be given $\ge 2$ marks Let questions be $a,\text{ }b,\text{ }c,\text{ }d,\text{ }e,\text{ }f,\text{ }g,\text{ }h$          and $a\text{ }+\text{ }b\text{ }+\text{ }c\text{ }+\text{ }d\text{ }+\text{ }e\text{ }+f+\text{ }g\text{ }+\text{ }h\text{ }=\text{ }30$ Let a $a={{a}_{1}}+2$ so, ${{a}_{1}}\ge 0$ $b={{a}_{2}}+2$so, ${{a}_{2}}\ge 0,......,{{a}_{g}}\ge 0$ So, $\left. \begin{matrix} {{a}_{1}}+{{a}_{2}}+......+{{a}_{8}} \\ +2+2+......+2 \\ \end{matrix} \right\}=30$ $\Rightarrow$ ${{a}_{1}}+{{a}_{2}}+.....+{{a}_{8}}=30-16=14$ So, this is a problem of distributing 14   articles in 8 groups.                    Number of ways $={{\,}^{14+8-1}}{{C}_{8-1}}={{\,}^{21}}{{C}_{7}}$