If the coefficients of $x$ and $x^{2}$ in the expansion of $(1+x)^{\mathrm{p}}(1-x)^{\mathrm{q}}, \mathrm{p}, \mathrm{q} \leq 15$, are $-3$ and $-5$ respectively, then the coefficient of $x^{3}$ is equal to _____________.
Answer (integer)
23
Solution
<p>Coefficient of x in ${(1 + x)^p}{(1 - x)^q}$</p>
<p>$- {}^p{C_0}\,{}^q{C_1} + {}^p{C_1}\,{}^q{C_0} = - 3 \Rightarrow p - q = - 3$</p>
<p>Coefficient of x<sup>2</sup> in ${(1 + x)^p}{(1 - x)^q}$</p>
<p>${}^p{C_0}\,{}^q{C_2} - {}^p{C_1}\,{}^q{C_1} + {}^p{C_2}\,{}^q{C_0} = - 5$</p>
<p>${{q(q - 1)} \over 2} - pq + {{p(q - 1)} \over 2} = - 5$</p>
<p>${{{q^2} - q} \over 2} - (q - 3)q + {{(q - 3)(q - 4)} \over 2} = - 5$</p>
<p>$\Rightarrow q = 11,\,p = 8$</p>
<p>Coefficient of x<sup>3</sup> in ${(1 + x)^8}{(1 - x)^{11}}$</p>
<p>$$ = - {}^{11}{C_3} + {}^8{C_1}\,{}^{11}{C_2} - {}^8{C_2}\,{}^{11}{C_1} + {}^8{C_3} = 23$$</p>
About this question
Subject: Mathematics · Chapter: Binomial Theorem · Topic: Binomial Expansion
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