In an examination the maximum marks for each of the three papers are $$50$$ each. Maximum marks for the fourth paper are $$100$$. The number of ways in which the candidate can score $$60$$% marks in aggregate is

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Infinity Learn · Accepted Answer

The candidate must score $$150$$ marks.∴ Required $$number=$$ coefficient of $$x150$$ in $$1+x+$$…$$+x5031+x+$$…$$+x100=$$ coefficient of $$x150$$ in $$1$$−$$x511$$−$$x31$$−$$x1011$$−$$x=$$ coefficient of $$x150$$ in $$1$$−$$x5131$$−$$x101(1$$−$$x)$$−$$4=$$ coefficient of $$x150$$ in $$1$$−$$3x51+3x102$$−$$x1531$$−$$x101(1$$−$$x)$$−$$4$$[leaving terms containing powers of x greater than $$150$$]$$=$$ coefficient of $$x150$$ in (1$$−$$x)−$$4$$−$$3$$. coefficient of $$x99$$ in (1$$−$$x)−$$4+3$$ coefficient of $$x48$$ in (1$$−$$x)−$$4$$ coefficient of $$x49$$ in (1$$−$$x)−$$4$$ $$=153C150$$−$$3$$⋅$$102C99+3$$⋅$$51C48$$−$$52C49$$.$$=153$$⋅$$152$$⋅$$1516$$−$$3$$⋅$$102$$⋅$$101$$⋅$$1006+3$$⋅$$51$$⋅$$50$$⋅$$496$$−$$52$$⋅$$51$$⋅$$506=$$ $$110556$$

In an examination the maximum marks for each of the three papers are 50 each. Maximum marks for the fourth paper are 100. The number of ways in which the candidate can score 60% marks in aggregate is

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