Question:
The coefficient of $x^5$ in the expansion of $(1 + x^2)^5(1 + x)^4$ is
The coefficient of $x^5$ in the expansion of $(1 + x^2)^5(1 + x)^4$ is
Updated On: Mar 4, 2024
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The Correct Option is B
Solution and Explanation
We have,
$\left(1+x^{2}\right)^{5}={ }^{5} C_{0}\left(x^{2}\right)^{0}+{ }^{5} C_{1}\left(x^{2}\right)^{1}+{ }^{5} C_{2}\left(x^{2}\right)^{2}$
$+{ }^{5} C_{3}\left(x^{2}\right)^{3}+{ }^{5} C_{4}\left(x^{2}\right)^{4}+{ }^{5} C_{5}\left(x^{2}\right)^{5} $
$= 1+5 x^{2}+10 x^{4}+10 x^{6}+5 x^{8}+x^{10} $
$(1+x)^{4}={ }^{4} C_{0} x^{0}+{ }^{4} C_{1} x^{1}+{ }^{4} C_{2} x^{2}+{ }^{4} C_{3} x^{3}+{ }^{4} C_{4} x^{4} $
$= 1+4 x+6 x^{2}+4 x^{3}+x^{4}$
$\therefore$ Coefficient of $x^{5}$ in the product of
$\left(1+x^{2}\right)^{5}(1+x)^{4} $
$=\left(5 x^{2}\right) \cdot\left(4 x^{3}\right)+\left(10 x^{4}\right).(4 x) $
$=20 x^{5}+40 x^{5} $
$=60\, x^{5}$
$\left(1+x^{2}\right)^{5}={ }^{5} C_{0}\left(x^{2}\right)^{0}+{ }^{5} C_{1}\left(x^{2}\right)^{1}+{ }^{5} C_{2}\left(x^{2}\right)^{2}$
$+{ }^{5} C_{3}\left(x^{2}\right)^{3}+{ }^{5} C_{4}\left(x^{2}\right)^{4}+{ }^{5} C_{5}\left(x^{2}\right)^{5} $
$= 1+5 x^{2}+10 x^{4}+10 x^{6}+5 x^{8}+x^{10} $
$(1+x)^{4}={ }^{4} C_{0} x^{0}+{ }^{4} C_{1} x^{1}+{ }^{4} C_{2} x^{2}+{ }^{4} C_{3} x^{3}+{ }^{4} C_{4} x^{4} $
$= 1+4 x+6 x^{2}+4 x^{3}+x^{4}$
$\therefore$ Coefficient of $x^{5}$ in the product of
$\left(1+x^{2}\right)^{5}(1+x)^{4} $
$=\left(5 x^{2}\right) \cdot\left(4 x^{3}\right)+\left(10 x^{4}\right).(4 x) $
$=20 x^{5}+40 x^{5} $
$=60\, x^{5}$
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Concepts Used:
Binomial Theorem
The binomial theorem formula is used in the expansion of any power of a binomial in the form of a series. The binomial theorem formula is
Properties of Binomial Theorem
- The number of coefficients in the binomial expansion of (x + y)n is equal to (n + 1).
- There are (n+1) terms in the expansion of (x+y)n.
- The first and the last terms are xn and yn respectively.
- From the beginning of the expansion, the powers of x, decrease from n up to 0, and the powers of a, increase from 0 up to n.
- The binomial coefficients in the expansion are arranged in an array, which is called Pascal's triangle. This pattern developed is summed up by the binomial theorem formula.