# Binomial Theorem: Greatest Coefficient of a binomial expansion

#### Finding the Greatest Coefficient of a binomial expansion:

To find the greatest coefficient of a binomial expansion of $$(a + b)^n$$, we need to find out the greatest value of $$k$$. As we know that the coefficient of the general term of the binomial expansion $$(a + b)^n$$ is $$\binom{n}{k}$$, where $$k = 0, 1, 2, 3.....n$$. Here $$n$$ could be an even number or odd number hence the greatest coefficient will depend on the value of $$n$$.

Case(1): If $$n$$ is an even number then for greatest coefficient $$k = \frac{n}{2}$$. Hence the greatest coefficient will be $$\binom{n}{\frac{n}{2}}$$ or we can also write it as $$nC_{\frac{n}{2}}$$.

Case(2): If $$n$$ is an odd number then for greatest coefficient $$k = \frac{(n - 1)}{2}$$ or $$k = \frac{(n + 1)}{2}$$. Hence the greatest coefiicient will be either $$\binom{n}{\frac{(n - 1)}{2}}$$ or $$\binom{n}{\frac{(n + 1)}{2}}$$. The final value for both will be same so we can take any one of them.

Example(1): Find the greatest coefficient of a binomial expansion $$(1 + 2x)^4$$?

Solution: Given, $$n = 4$$, $$a = 1$$, $$b = 2x$$

Here the value of $$n$$ is an even number, hence $$k = \frac{n}{2}$$ $$k = \frac{4}{2}$$ $$k = 2$$ The greatest coefficient of the binomial expansion. $$= \binom{n}{\frac{n}{2}}$$ $$= \binom{4}{2}$$ This is the greatest coefficient of the binomial expansion of $$(1 + 2x)^4$$. We can further solve $$\binom{4}{2}$$ to get the final value. $$\binom{4}{2} = \frac{4!}{(4 - 2)! \ 2!}$$ $$= \frac{4 \times 3 \times 2!}{2! \ 2!}$$ $$= \frac{4 \times 3}{2 \times 1}$$ $$\binom{4}{2} = 6$$

Example(2): Find the greatest coefficient of a binomial expansion $$(2x + 3y)^{12}$$?

Solution: Given, $$n = 12$$, $$a = 2x$$, $$b = 3y$$

Here the value of $$n$$ is an even number, hence $$k = \frac{n}{2}$$ $$k = \frac{12}{2}$$ $$k = 6$$ The greatest coefficient of the binomial expansion. $$= \binom{n}{\frac{n}{2}}$$ $$= \binom{12}{6}$$ This is the greatest coefficient of the binomial expansion of $$(2x + 3y)^{12}$$. We can further solve $$\binom{12}{6}$$ to get the final value. $$\binom{12}{6} = \frac{12!}{(12 - 6)! \ 6!}$$ $$= \frac{12 \times 11 \times 10 \times 9 \times 8 \times 7}{6!}$$ $$= \frac{12 \times 11 \times 10 \times 9 \times 8 \times 7}{6 \times 5 \times 4 \times 3 \times 2 \times 1}$$ $$\binom{12}{6} = 924$$

Example(3): Find the greatest coefficient of a binomial expansion $$(2x + 3y)^7$$?

Solution: Given, $$n = 7$$, $$a = 2x$$, $$b = 3y$$

Here the value of $$n$$ is an odd number, hence the value of $$k$$ will be either $$k = \frac{n - 1}{2}$$ or $$k = \frac{n - 1}{2}$$. $$k = \frac{n - 1}{2}$$ $$k = \frac{7 - 1}{2}$$ $$k = \frac{6}{2}$$ $$k = 3$$ The greatest coefficient of the binomial expansion. $$= \binom{n}{\frac{n - 1}{2}}$$ $$= \binom{7}{3}$$ This is the greatest coefficient of the binomial expansion of $$(2x + 3y)^7$$. We can further solve $$\binom{7}{3}$$ to get the final value. $$\binom{7}{3} = \frac{7!}{(7 - 3)! \ 3!}$$ $$= \frac{7 \times 6 \times 5 \times 4!}{4! \ 3!}$$ $$= \frac{7 \times 6 \times 5}{3!}$$ $$= \frac{7 \times 6 \times 5}{3 \times 2 \times 1}$$ $$\binom{7}{3} = 35$$