Coefficient of $x^{50}$ in ${(1 + x)}^{1001} - x^{1001}$

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Solution

$(1 + x)^{1001} - x^{1001}$
$= 1 +^{1001} C_{1} x +^{1001} C_{2} x^{2} + . . . . . . . . . . . +^{1001} C_{50} x^{50} + . . . . . . . +^{1001} C_{1001} x^{1001} - x^{1001}$
Hence the coefficient of $x^{50} =^{1001} C_{50}$

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