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

The candidate must score 150 marks.

∴ Required number

$\begin{array}{c}=\text{ coefficient of }{\mathrm{x}}^{150}\text{ in }{\left(1+\mathrm{x}+\dots +{\mathrm{x}}^{50}\right)}^3\left(1+\mathrm{x}+\dots +{\mathrm{x}}^{100}\right)\\ =\text{ coefficient of }{\mathrm{x}}^{150}\text{ in }{\left(\frac{1-{\mathrm{x}}^{51}}{1-\mathrm{x}}\right)}^3\frac{1-{\mathrm{x}}^{101}}{1-\mathrm{x}}\\ =\text{ coefficient of }{\mathrm{x}}^{150}\text{ in }{\left(1-{\mathrm{x}}^{51}\right)}^3\left(1-{\mathrm{x}}^{101}\right)(1-\mathrm{x}{)}^{-4}\\ =\text{ coefficient of }{\mathrm{x}}^{150}\text{ in }\left(1-3{\mathrm{x}}^{51}+3{\mathrm{x}}^{102}-{\mathrm{x}}^{153}\right)\left(1-{\mathrm{x}}^{101}\right)(1-\mathrm{x}{)}^{-4}\end{array}$

[leaving terms containing powers of x greater than 150]

$$\begin{gathered} =\text{ coefficient of }{\mathrm{x}}^{150}\text{ in }(1-\mathrm{x}{)}^{-4}-3\text{. coefficient of }{\mathrm{x}}^{99}\text{ in }(1-\mathrm{x}{)}^{-4}+3\text{ coefficient of }{\mathrm{x}}^{48}\text{ in }(1-\mathrm{x}{)}^{-4}\text{ coefficient of }{\mathrm{x}}^{49}\text{ in }(1-\mathrm{x}{)}^{-4} \\ \begin{array}{c}{=}^{153}{\mathrm{C}}_{150}-3{\cdot }^{102}{\mathrm{C}}_{99}+3{\cdot }^{51}{\mathrm{C}}_{48}{-}^{52}{\mathrm{C}}_{49}.\\ =\frac{153\cdot 152\cdot 151}{6}-3\cdot \frac{102\cdot 101\cdot 100}{6}+3\cdot \frac{51\cdot 50\cdot 49}{6}-\frac{52\cdot 51\cdot 50}{6}\\ = 110556\end{array} \end{gathered}$$