Question

If the sum of binomial coefficients in the expansion ${({\text{1 + x)}}^{\text{n}}}$ is 1024, then what is the largest coefficient in expansion?

Answer 1

Hint: The largest term in the expansion of ${({\text{1 + x)}}^{\text{n}}}$ is the middle term of the expansion. We will find the sum of binomial coefficients and then find the middle term to get the value of largest coefficient.



Complete step-by-step answer:
Now, the expansion of ${({\text{1 + x)}}^{\text{n}}}$ can be written as,
${({\text{1 + x)}}^{\text{n}}}$ = $^{\text{n}}{{\text{C}}_0}{\text{ + }}{}^{\text{n}}{{\text{C}}_1}{{\text{x}}^1}{\text{ + }}{}^{\text{n}}{{\text{C}}_2}{{\text{x}}^2}{\text{ + }}................{\text{ + }}{}^{\text{n}}{{\text{C}}_{\text{n}}}{{\text{x}}^{\text{n}}}$
Now, to find the sum of coefficients we put x = 1 in the expansion of ${({\text{1 + x)}}^{\text{n}}}$. Therefore,
${(1{\text{ + 1)}}^{\text{n}}}{\text{ = }}{{\text{ }}^{\text{n}}}{{\text{C}}_0}{\text{ + }}{}^{\text{n}}{{\text{C}}_1}{\text{ + }}{}^{\text{n}}{{\text{C}}_2}{\text{ + }}................{\text{ + }}{}^{\text{n}}{{\text{C}}_{\text{n}}}$
${2^{\text{n}}}$ = $^{\text{n}}{{\text{C}}_0}{\text{ + }}{}^{\text{n}}{{\text{C}}_1}{\text{ + }}{}^{\text{n}}{{\text{C}}_2}{\text{ + }}................{\text{ + }}{}^{\text{n}}{{\text{C}}_{\text{n}}}$
So, the sum of coefficients of the given expansion = ${2^{\text{n}}}$
According to question, the sum of coefficient = 1024
Therefore,
${2^{\text{n}}}$ = 1024
So, n = 10
So, the expansion becomes ${({\text{1 + x)}}^{10}}$.
Now, we have to find the largest coefficient. Now, every expansion gives a number of terms which is one plus the index of the expansion i.e. if there is an expansion of index n, then by expanding the expansion, we get (n + 1) terms.
So, to find the middle term it is important that n is even or odd. If n is even then the total number of terms are n + 1. So, the middle term is $\dfrac{{({\text{n + 2)}}}}{2}$th term.
If n is odd then the middle terms are $\dfrac{{({\text{n + 1)}}}}{2}$th and ($\dfrac{{({\text{n + 1)}}}}{2}$ + 1) th term.
Now, given expansion has even index so, its middle term is $\dfrac{{({\text{n + 2)}}}}{2}$th term, i.e. 6^th term is middle term of the expansion ${({\text{1 + x)}}^{10}}$.
Now, rth term in the expansion ${({\text{1 + x)}}^{\text{n}}}$ can be written as T_r+1 = ${}^{\text{n}}{{\text{C}}_{\text{r}}}{{\text{x}}^{\text{r}}}$.
So, the coefficient of 6^th term = ${}^{10}{{\text{C}}_5}$ = 252.
So, the largest coefficient is 252.

Note: To solve such types of questions, we will first find the value of expansion. If the value of the index is not given, then we have to find the value of the index by applying the condition given in the question. The largest term of any expansion is the middle term. So, it is important that we find the correct middle term. Most of the students make mistakes in such questions while finding the middle term. Even index has only one middle term and odd index has two middle terms each of which is the largest.