In how many ways in which an examiner can assign 30 marks to 8 questions, giving not less than 2 marks to any question.

Explanation

If examiner given marks any seven questions 2 (each) marks, then marks on remaining questions given by examiner=`-7xx2+30=16`
IF `x_(i)` are the marks assigned to `i`th questions, then
`x_(1)+x_(2)+x_(3)+ . . .+x_(8)=30 and 2lex_(i)le16`
for `i=1,2,3,. . ..,8`
Here, `l=2,m=16,r=8 and n=30`
`therefore`Required number of ways
=Coefficient of `x^(30-2xx8)` in the expansion of
`(1-x^(16-2+1)^(8)(1-x)^(-8)`
=Coefficient of `x^(14)` in the expansion of
`(1-x^(15))^(8)(1+.^(8)C_(1)x+.^(9)C_(2)x^(2)+ . ..+.^(21)C_(14)x^(14)+ . . .)`
=Coefficient of `x^(14)` in the expansion of
`(1+.^(8)C_(1)x+.^(9)C_(2)x^(2)+ . . .+.^(21)C_(14)x^(14)+ . ..)`
`=.^(21)C_(14)=.^(21)C_(7)`