The coefficients of \(x^{50}\) in \((1+x)^{101}\) \(\left(1-x+x^2\right)^{100}\) is......
The coefficients of \(x^{50}\) in \((1+x)^{101}\) \(\left(1-x+x^2\right)^{100}\) is......
1
-1
0
2
Solution
Given expression is
\(\begin{aligned}
& (1+x)^{101}\left(1-x+x^2\right)^{100} \\
& =(1+x)\left[(1+x)\left(1-x+x^2\right)\right]^{100} \\
& =(1+x)\left(1+x^3\right)^{100}
\end{aligned}\)
So, coefficient of \(x^{50}\) in \((1+x)^{101}\left(1-x+x^2\right)^{100}\) \(=\) coefficient of \(x^{50}\) in \((1+x)\left(1+x^3\right)^{100}\)
\(=\) coefficient of \(x^{50}\) in \(\left(1+x^3\right)^{100}\)
+ coefficient of \(x^{49}\) in \(\left(1+x^3\right)^{100}\)
\(\because\) In the expansion of \(\left(1+x^3\right)^{100}\), the power of \(x\) is multiple of 3 , and while 50 and 49 are not multiple of 3.
So, coefficient of \(x^{50}\) and 49 in the expansion of \(\left(1+x^3\right)^{100}\) is zero.
\(\therefore\) Coefficient of \(x^{50}\) in \((1+x)^{100}\left(1-x+x^2\right)^{101}\) is zero.
Hence, option (c) is correct.