Question

The largest coefficient in the expansion of \[{\left( {1 + x} \right)^{24}}\;\] is
A. \[{}^{24}{C_{24}}\]
B. \[{}^{24}{C_{13}}\]
C. \[{}^{24}{C_{12}}\]
D. \[{}^{24}{C_{11}}\]

Answer 1

Hint: In this question, we need to find the largest coefficient in the expansion of \[{\left( {1 + x} \right)^{24}}\;\]. For this, we will use the below-mentioned formula and concept of binomial coefficient to get the desired result.

Formula used: The following formula will be used to solve the given question.
We know that the largest coefficient is the binomial coefficient of the middle term.
The largest coefficient is \[{}^n{C_{\dfrac{n}{2}}}\] if n is even.

Complete step-by-step solution:
We know that \[{\left( {1 + x} \right)^{24}}\;\]
Let us find the largest coefficient in the above expansion.
Here, \[{\left( {1 + x} \right)^{24}}\;\] is in the form of \[{\left( {1 + x} \right)^n}\;\].
Thus, by comparing \[{\left( {1 + x} \right)^{24}}\;\] with \[{\left( {1 + x} \right)^n}\;\], we get
\[n = 24\]
Here, we can say that n is even.
So, the largest coefficient is the binomial coefficient of the middle term.
Thus, the largest coefficient is \[{}^n{C_{\dfrac{n}{2}}}\].
Now, put \[n = 24\] in \[{}^n{C_{\dfrac{n}{2}}}\].
Hence, we get \[{}^{24}{C_{\dfrac{{24}}{2}}}\]
So, the largest coefficient is \[{}^{24}{C_{12}}\]
Thus, the largest coefficient in the expansion of \[{\left( {1 + x} \right)^{24}}\;\] is \[{}^{24}{C_{12}}\].
Therefore, the correct option is (C).
Additional information: A binomial coefficient is the variety of available combinations of r objects from a group of n objects. It is also an input in Pascal's triangle. Since they are coefficients in the binomial theorem, these numbers are known as binomial coefficients. If n is odd in the given expansion then the largest coefficient will be \[{}^n{C_{\dfrac{{n - 1}}{2}}}\] and \[{}^n{C_{\dfrac{{n + 1}}{2}}}\].

Note: Many students generally make mistakes in writing the formula for the largest coefficient in the given expansion if n is even. Thus, end result may get wrong. Here, to find the value of n it is necessary to compare the value of n in the given expansion with the standard expansion.