Try this problem from I.S.I. B.Stat Entrance Objective Problem based on Binomial Expression.
The coefficient of \(x^{2}\) in the binomial expression of \((1+x+x^{2})^{10}\) is
Cefficient
Binomials
Combinations
Answer:\({10 \choose 1}+{10 \choose 2}\).
B.Stat Objective Problem 117
Challenges and Thrills of Pre-College Mathematics by University Press
\((\frac{1-x^{3}}{1-x})^{10}\)
=\((1-x^{3})^{10}(1-x)^{-10}\)
=\((1-{10 \choose 1}x^{3}\)
\(+{10 \choose 2}x^{6}+....)(1-10(-x)\)
\(+\frac{(-10)(-10)-1}{2}(-x)^{2}+.....)\)
coefficient of \(x^{2}\)=\(\frac{(-10)((-10)-1)}{2}\)
=\({10 \choose 1}+{10 \choose 2}\).
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