Question

In the expansion of ${\left( {1 + 3x + 2{x^2}} \right)^6}$ the coefficient of ${x^{11}}$ is
(A)144
(B)288
(C)216
(D)576

Answer 1

Hint: This question is done with the help of the concept of Binomial expansion. In this expansion,
${\left( {1 + x} \right)^n} = 1 + {}^n{C_1}x + {}^n{C_2}{x^2} + {}^n{C_3}{x^3} + $_ _ _$ + {}^n{C_n}{x^n}$
Here we have to compare these terms with the question we are given with. Here ‘$C$’ denotes the ‘Combination’’. It is easy to remember binomials a ‘Bi’ means two and a binomial will have two terms. The binomial theorem is a result of expanding the powers of binomials or sum of two terms. The coefficient of the terms in the expansion are the binomial coefficient.

Complete step by step solution:
Given equation,${\left( {1 + 3x + 2{x^2}} \right)^6}$
On making some changes this equation can be written as ${\left( {1 + x\left( {3 + 2x} \right)} \right)^6}$
Now, if we apply binomial expansion on this equation according to the formula we get,
${\left( {1 + x\left( {3 + 2x} \right)} \right)^6}$$ = 1 + {}^6{C_1}x\left( {3 + 2x} \right) + {}^6{C_2}{x^2}{\left( {3 + 2x} \right)^2}$$ + {}^6{C_3}{x^3}{\left( {3 + 2x} \right)^3} + {}^6{C_4}{x^4}{\left( {3 + 2x} \right)^4} + $
                                  ${}^6{C_5}{x^5}{\left( {3 + 2x} \right)^5} + {}^6{C_6}{x^6}{\left( {3 + 2x} \right)^6}$
We know that we have to get the coefficient of ${x^{11}}$
We can get the term ${x^{11}}$ from the above expansion on comparing. As we know we can get this term if ${x^6}$multiplies with${x^5}$. We already have ${x^6}$term in our equation. Now we have to find ${x^5}$which we can get from${\left( {3 + 2x} \right)^6}$. So on expansion
Expand, $ \to {\left( {3 + 2x} \right)^6}$and we have to get the term having ${x^5}$
So that term can be obtained as,
$ = {}^6{C_5} \times 3 \times {\left( {2x} \right)^5}$
Now we got both terms ${x^5}$and${x^6}$. On multiplying both we will get ${x^{11}}$
And when we multiply this term ${x^5}$with the term ${x^6}$it will give ${x^{11}}$in the term ${}^6{C_6}{x^6}{\left( {3 + 2x} \right)^6}$
                   $ = {}^6{C_6}{x^6} \times {}^6{C_5} \times 3 \times {\left( {2x} \right)^5}$
On solving this we get,
                   $
   = 1 \times 6 \times 3 \times {2^5} \times {x^{11}} \\
   = 576{x^{11}} \\
 $
So, the coefficient of ${x^{11}}$is$576$.
So option (D) is the correct option.

Note: We have to multiply ${x^6}$with ${x^5}$to get${x^{11}}$. Apply the combination properly because mistakes can be made there. Use the bracket when there is power to a function so that the calculation becomes easier.