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Mathematics Question on binomial expansion formula

For the expression (1−x)100(1-x)^{100}. Then sum of coefficient of first 50 terms is:

A

99C49^{99}C_{49}

B

−100C502-\frac{^{100}C_{50}}{2}

C

−99C49-^{99}C_{49}

D

−101C50-^{101}C_{50}

Answer

−100C502-\frac{^{100}C_{50}}{2}

Explanation

Solution

Sum of coefficient of first 50 terms

(t)(t) =100C0−100C1+...+100C40= ^{100}C_0-^{100}C_1+...+^{100}C_{40}

Now

100C0−100C1+...+100C100=0^{100}C_0-^{100}C_1+...+^{100}C_{100}=0

2[100C0−100C1+...]+100C50=02[^{100}C_0-^{100}C_1+...]+ ^{100}C_{50}=0

∴  t=−12100C50\therefore \; t= -\frac{1}{2}^{100}C_{50}

The correct option is (B): −100C502-\frac{^{100}C_{50}}{2}