Question

The coefficient of ${{x}^{7}}$ in ${{\left( 1-x-{{x}^{2}}+{{x}^{3}} \right)}^{6}}$ is

& A.-144 \\\ & B.144 \\\ & C.-128 \\\ & D.-142 \\\ \end{aligned}$$

Answer 1

In this question, we are given a polynomial and we need to find a coefficient of ${{x}^{7}}$. For this, we first need to simplify the polynomial. We will factorise the polynomial. Then, we will find all the possible combinations of powers of x from the factor which when multiplied will give us the power of x as 7. Then, we will find the coefficients of all these combinations and add. For finding the coefficient of any ${{x}^{n}}$ in ${{\left( 1+x \right)}^{p}}$ we use formula: ${}^{p}{{C}_{n}}=\dfrac{p!}{n!\left( p-n \right)!}$ and for coefficient of any ${{x}^{n}}{{\left( 1-x \right)}^{p}}$ we use formula ${{\left( -1 \right)}^{n}}{}^{p}{{C}_{n}}={{\left( -1 \right)}^{n}}\dfrac{p!}{n!\left( p-n \right)!}$.

Complete step-by-step solution
Here, we are given the polynomial as ${{\left( 1-x-{{x}^{2}}+{{x}^{3}} \right)}^{6}}$.
Let us factorise it $\left( 1-x-{{x}^{2}}+{{x}^{3}} \right)$. Taking $-{{x}^{2}}$ common from third and fourth term we get: $\left( \left( 1-x \right)-{{x}^{2}}\left( 1-x \right) \right)$. Now take the common (1-x) common we get: $\left( 1-x \right)\left( 1-{{x}^{2}} \right)$. As we know that, ${{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right)$ so we get: $\left( 1-x \right)\left( 1-x \right)\left( 1+x \right)\Rightarrow {{\left( 1-x \right)}^{2}}\left( 1+x \right)$.
Now, ${{\left( 1-x-{{x}^{2}}+{{x}^{3}} \right)}^{6}}$ becomes ${{\left( {{\left( 1-x \right)}^{2}}\left( 1+x \right) \right)}^{6}}\Rightarrow {{\left( 1-x \right)}^{12}}{{\left( 1+x \right)}^{6}}$.
Now, let us find combination of ${{x}^{m}}$ from ${{\left( 1-x \right)}^{12}}$ and ${{x}^{n}}$ from ${{\left( 1+x \right)}^{6}}$ such that ${{x}^{m}}\cdot {{x}^{n}}={{x}^{m+n}}={{x}^{7}}$ and then apply following formula to find coefficient.
Coefficient of ${{x}^{m}}$ in ${{\left( 1+x \right)}^{p}}={}^{p}{{C}_{m}}=\dfrac{p!}{m!\left( p-m \right)!}$.
Coefficient of ${{x}^{n}}$ in ${{\left( 1-x \right)}^{p}}={}^{p}{{C}_{n}}=\dfrac{p!}{n!\left( p-n \right)!}$.
Combination and their coefficient are:
1. Coefficient of x in ${{\left( 1-x \right)}^{12}}\times$ coefficient of ${{x}^{6}}$ in ${{\left( 1+x \right)}^{6}}$.

& \Rightarrow \left( -1 \right){}^{12}{{C}_{1}}\times {}^{6}{{C}_{6}} \\\ & \Rightarrow \dfrac{-12!}{1!\left( 12-1 \right)!}\times \dfrac{6!}{6!\left( 6-6 \right)!} \\\ & \Rightarrow \dfrac{-12\times 11!}{11!}\times \dfrac{1}{0!} \\\ & \Rightarrow -12\times 1=-12 \\\ \end{aligned}$$ 2\. Coefficient of ${{x}^{2}}$ in ${{\left( 1-x \right)}^{12}}\times $ coefficient of ${{x}^{5}}$ in ${{\left( 1+x \right)}^{6}}$. $$\begin{aligned} & \Rightarrow {{\left( -1 \right)}^{2}}{}^{12}{{C}_{2}}\times {}^{6}{{C}_{5}} \\\ & \Rightarrow \dfrac{12!}{2!10!}\times \dfrac{6!}{5!} \\\ & \Rightarrow \dfrac{12\times 11\times 10!}{2\times 10!}\times \dfrac{6\times 5!}{5!} \\\ & \Rightarrow 66\times 6 \\\ & \Rightarrow 396 \\\ \end{aligned}$$ 3\. Coefficient of ${{x}^{3}}$ in ${{\left( 1-x \right)}^{12}}\times $ coefficient of ${{x}^{4}}$ in ${{\left( 1+x \right)}^{6}}$. $$\begin{aligned} & \Rightarrow {{\left( -1 \right)}^{3}}{}^{12}{{C}_{3}}\times {}^{6}{{C}_{4}} \\\ & \Rightarrow \dfrac{-12!}{3!9!}\times \dfrac{6!}{4\times 2!} \\\ & \Rightarrow \dfrac{-12\times 11\times 10\times 9!}{6\times 9!}\times \dfrac{6\times 5\times 4!}{4!\times 2} \\\ & \Rightarrow -220\times 15 \\\ & \Rightarrow -3300 \\\ \end{aligned}$$ 4\. Coefficient of ${{x}^{4}}$ in ${{\left( 1-x \right)}^{12}}\times $ coefficient of ${{x}^{3}}$ in ${{\left( 1+x \right)}^{6}}$. $$\begin{aligned} & \Rightarrow {{\left( -1 \right)}^{4}}{}^{12}{{C}_{4}}\times {}^{6}{{C}_{3}} \\\ & \Rightarrow \dfrac{12!}{4!8!}\times \dfrac{6!}{3!3!} \\\ & \Rightarrow \dfrac{12\times 11\times 10\times 9\times 8!}{24\times 8!}\times \dfrac{6\times 5\times 4\times 3!}{6\times 3!} \\\ & \Rightarrow 495\times 20 \\\ & \Rightarrow 9900 \\\ \end{aligned}$$ 5\. Coefficient of ${{x}^{5}}$ in ${{\left( 1-x \right)}^{12}}\times $ coefficient of ${{x}^{2}}$ in ${{\left( 1+x \right)}^{6}}$. $$\begin{aligned} & \Rightarrow {{\left( -1 \right)}^{5}}{}^{12}{{C}_{5}}\times {}^{6}{{C}_{2}} \\\ & \Rightarrow \dfrac{-12!}{5!7!}\times \dfrac{6!}{2!4!} \\\ & \Rightarrow \dfrac{-12\times 11\times 10\times 9\times 8\times 7!}{5\times 4\times 3\times 2\times 7!}\times \dfrac{6\times 5\times 4!}{4!\times 2} \\\ & \Rightarrow -792\times 15 \\\ & \Rightarrow -11880 \\\ \end{aligned}$$ 6\. Coefficient of ${{x}^{6}}$ in ${{\left( 1-x \right)}^{12}}\times $ coefficient of x in ${{\left( 1+x \right)}^{6}}$. $$\begin{aligned} & \Rightarrow {{\left( -1 \right)}^{6}}{}^{12}{{C}_{6}}\times {}^{6}{{C}_{1}} \\\ & \Rightarrow \dfrac{12!}{6!6!}\times \dfrac{6!}{1!5!} \\\ & \Rightarrow 924\times 6 \\\ & \Rightarrow 5544 \\\ \end{aligned}$$ 7\. Coefficient of ${{x}^{7}}$ in ${{\left( 1-x \right)}^{12}}\times $ coefficient of ${{x}^{0}}$ in ${{\left( 1+x \right)}^{6}}$. $$\begin{aligned} & \Rightarrow {{\left( -1 \right)}^{7}}{}^{12}{{C}_{7}}\times {}^{6}{{C}_{0}} \\\ & \Rightarrow \dfrac{-12!}{7!6!}\times \dfrac{6!}{0!6!} \\\ & \Rightarrow -792\times 1 \\\ & \Rightarrow -792 \\\ \end{aligned}$$ Now, adding all these combinations will give us a coefficient of ${{x}^{7}}$. Hence, the coefficient of ${{x}^{7}}$ is $$\Rightarrow -12+396-3300+9900-11880+5544-792=-144$$. Therefore, coefficient of ${{x}^{7}}$ in ${{\left( 1-x-{{x}^{2}}+{{x}^{3}} \right)}^{6}}$ is -144. **Therefore, option A is the correct answer.** **Note:** Students should take care while evaluating ${}^{n}{{C}_{r}}$. We are dealing with huge numbers here so take care of positive and negative signs. Students should not miss any combination and don't forget to take ${{\left( -1 \right)}^{n}}$ for coefficient of ${{x}^{n}}$ in ${{\left( 1-x \right)}^{p}}$. To ease calculation students should know that ${}^{n}{{C}_{r}}={}^{n}{{C}_{n-r}}$.