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In an examination, the maximum mark for ...

In an examination, the maximum mark for each of the three papers is 50 and the maximum mark for the fourth paper is 100. Find the number of ways in which the candidate can score 605 marks in aggregate.

Text Solution

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Let the marks scored by the candidate in four papers be `x_(1),x_(2),x_(3) " and " x_(4)`.
Then `x_(1)+x_(2)+x_(3)+x_(4)=150 (60% " of " 250=150)`
Here, `0 le x_(1),x_(2),x_(3) le 50 " and " 0 le x_(4) le 100`.
`therefore` Number of ways candidate can score 60 % marks
= coefficient of `p^(150) " in " (1+p+p^(2)+..+p^(50))^(3)(1+p+p^(2)+..+p^(100))`
(In each bracket series is not extended to infinite terms as upper limit of each variable is less than 150)
=coefficient of `p^(150) " in " ((1-p^(51))/(1-p))^(3)((1-p^(101))/(1-p))`
=coefficient of `p^(150) " in " (1-p^(51))^(3)(1-p^(101))(1-p)^(-4)`
=coefficient of `p^(150) " in " (1-3p^(51)+3p^(102)-p^(101))xx(1+ ""^(4)C_(1)p+ ""^(5)C_(2)p^(2)+ ""^(6)C_(3)p^(3)++..)`
`=""^(153)C_(3)-3xx""^(102)C_(3)+3xx ""^(51)C_(3)- ""^(52)C_(3)`
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