Question

The coefficient of ${\mathrm{x}}^{20}$ in the expansion of ${(1+3\mathrm{x}+3{\mathrm{x}}^2+{\mathrm{x}}^3)}^{20}$ is

• A. ${}^{60}{\mathrm{C}}_{40}$
• B. ${}^{30}{C}_{20}$
• C. ${}^{15}{C}_2$
• D. None of these

Answer 1

The correct option is A

${}^{60}{\mathrm{C}}_{40}$

Explanation for the correct option:

Step-1: Simplification

Given, the expression:${(1+3\mathrm{x}+3{\mathrm{x}}^2+{\mathrm{x}}^3)}^{20}$.

Now, we know that ${\left(1+x\right)}^3=1+3x+3x^2+x^3$.

So, re-writing the given expression, we get:

$$\begin{gathered} {(1+3x+3x^2+x^3)}^{20}={\left[{\left(1+x\right)}^3\right]}^{20} \\ ={\left(1+x\right)}^{60} \end{gathered}$$

Therefore, we have to find the coefficient of ${\mathrm{x}}^{20}$ in the expression ${\left(1+x\right)}^{60}$

Step-2: Finding the coefficient of ${\mathrm{x}}^{20}$.

Formula to be used: We know that the coefficient of ${\mathrm{x}}^r$ in the binomial expansion of ${\left(1+x\right)}^n$is ${}^{n}C_r$.

Here, $n=60,r=20$.

So, the coefficient of ${\mathrm{x}}^{20}$ in the binomial expansion of ${\left(1+x\right)}^{60}$ is ${}^{60}C_{20}$.

Now, we know that ${}^{n}C_r={}^{n}C_{n-r}$.

So, ${}^{60}C_{20}={}^{60}C_{60-20}={}^{60}C_{40}$.

Hence, option (A) is the correct answer.