BITSAT 2014
Exam BITSAT
Year 2014
Subject Mathematics
Chapter Binomial Theorem BITSAT 2014 · Mathematics · Binomial Theorem
The coefficient of ^ 4 in the expansion of (1+x+x^ 2 +x^ 3 )^ 11 ,...
The coefficient of
\mathrmx 4 \mathrmx^{4} \mathrmx 4 in the expansion of
( 1 + x + x 2 + x 3 ) 11 , \left(1+x+x^{2}+x^{3}\right)^{11}, ( 1 + x + x 2 + x 3 ) 11 , is
Step-by-step solution
We have coefficient of
x 4 i n ( 1 + x + x 2 + x 3 ) 11 x^{4} in \left(1+x+x^{2}+x^{3}\right)^{11} x 4 in ( 1 + x + x 2 + x 3 ) 11 = coefficient of x 4 in ( 1 + x 2 ) 11 ( 1 + x ) 11 = coefficient of x 4 in ( 1 + x ) 11 + coefficient of x 2 in 11. ( 1 + x ) 11 + constant term is 11 C 2 ⋅ ( 1 + x ) 11 = 11 C 4 + 11. 11 C 2 + 11 C 2 = 990 \begin{array}{l}
=\text{ coefficient of } \mathrm{x}^{4} \text{ in }\left(1+\mathrm{x}^{2}\right)^{11}(1+\mathrm{x})^{11}\\
=\text{ coefficient of } \mathrm{x}^{4} \text{ in }(1+\mathrm{x})^{11}+\text{ coefficient }\\
\text{ of } \mathrm{x}^{2} \text{ in }\\
\text{ 11. }(1+\mathrm{x})^{11}+\text{ constant term is }{ }^{11} \mathrm{C}_{2} \cdot(1+\mathrm{x})^{11}\\
={ }^{11} \mathrm{C}_{4}+11 .{ }^{11} \mathrm{C}_{2}+{ }^{11} \mathrm{C}_{2}=990
\end{array} = coefficient of x 4 in ( 1 + x 2 ) 11 ( 1 + x ) 11 = coefficient of x 4 in ( 1 + x ) 11 + coefficient of x 2 in 11. ( 1 + x ) 11 + constant term is 11 C 2 ⋅ ( 1 + x ) 11 = 11 C 4 + 11. 11 C 2 + 11 C 2 = 990
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