Question

The sum of binomial coefficient in the expansion of \[{\left( {x + \dfrac{1}{x}} \right)^n}\] is 64, the term independent of x is equal to
A) 10
B) 20
C) 40
D) 60

Answer 1

Hint: Since the given sum uses a binomial expansion we will find the sum of its binomial coefficients and then the term independent of x.

Complete step-by-step answer:
Using binomial theorem,
\[{(x + a)^n} = n{C_0}{x^n} + n{C_1}{x^{n - 1}}a + n{C_2}{x^{n - 2}}{a^2} + .......n{C_n}{a^n}\]
Here the binomial coefficients are \[n{C_0}\], \[n{C_1}\], \[n{C_2}\]…….
Sum of these binomial coefficients is =\[\sum\nolimits_{r = 0}^n {n{C_r}} = {2^n}\]
\[
   \Rightarrow {2^n} = 64 \\
   \Rightarrow n = 8 \\
\]
Now the term independent of x is ,
\[{T_{r + 1}} = n{C_r}{(x)^{n - r}}{\left( {\dfrac{1}{x}} \right)^r}\]
\[ \Rightarrow {T_{r + 1}} = 6{C_r}{(x)^{6 - r}}{\left( {\dfrac{1}{x}} \right)^r}\]
\[
   \Rightarrow {T_{r + 1}} = 6{C_r}{x^{6 - r - r}} \\
   \Rightarrow {T_{r + 1}} = 6{C_r}{x^{6 - 2r}} \\
\]
Since the term is independent of x,
\[
  6 - 2r = 0 \\
  6 = 2r \\
  r = 3 \\
\]
Since r=3 then term independent of x will be,
\[
   \Rightarrow {T_{3 + 1}} = 6{C_3} \\
   \Rightarrow {T_4} = \dfrac{{6!}}{{3!\left( {6 - 3} \right)!}} \\
   \Rightarrow {T_4} = \dfrac{{6!}}{{3!3!}} \\
   \Rightarrow {T_4} = \dfrac{{6 \times 5 \times 4}}{{3 \times 2 \times 1}} \\
   \Rightarrow {T_4} = 20 \\
\]
So option B is the correct answer.

Additional information:
If n is any positive integer, then
  \[\begin{gathered}
  {(x + a)^n} = n{C_0}{x^n} + n{C_1}{x^{n - 1}}a + n{C_2}{x^{n - 2}}{a^2} + .......n{C_n}{a^n} \\
  {(x + a)^n} = \sum\nolimits_{r = 0}^n {n{C_r}{x^{n - r}}{a^r}} \\
\end{gathered} \]
 This is called a binomial theorem.
Here \[n{C_0}\],\[n{C_1}\],\[n{C_2}\]…….\[n{C_n}\] are binomial coefficients.
Where \[n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}\] \[0 < r < n\].
 Total number of terms in the expansion of \[{(x + a)^n}\] is n+1.

Note: We have to find a term independent of x that is \[{T_{r + 1}}\].
Since the term independent of x so power of x is 0.
The sum of the indices of x and a in each term is n.
Applications of binomial theorem:
Binomial theorem is used in economic prediction. This helps economists to predict that the economy will fall or bounce.
It is also used in architecture or civil engineering to predict the estimates of cost and time required for that project.
Weather forecasting is another field in which binomial theorem is used.