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 :

Solution :
30 marks to be alloted to 8 questions. Each question has to be given $$ \geqslant 2$$  marks
Let questions be $$a, b, c, d, e, f, g, h$$
and $$a + b + c + d + e + f + g + h = 30$$
Let, $$a = {a_1} + 2{\text{ so, }}{a_1} \geqslant 0$$
$$\eqalign{ & b = {a_2} + 2{\text{ so, }}{a_2} \geqslant 0,.....,{a_8} \geqslant 0 \cr & {\text{So, }}\left. {{a_1} + {a_2} + ..... + {a_8} + 2 + 2 + ..... + 2} \right\} = 30 \cr & \Rightarrow {a_1} + {a_2} + ..... + {a_8} = 30 - 16 = 14. \cr} $$
So, this is a problem of distributing 14 articles in 8 groups.
Number of ways $$ = {\,^{14 + 8 - 1}}{C_{8 - 1}} = {\,^{21}}{C_7}$$