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. $$ \binom{7}{3} = \frac{7!}{(7 - 3)! \ 3!} $$ $$ = \frac{7 \times 6 \times 5 \times 4!}{4! \ 3!} $$ $$ = \frac{7 \times 6 \times 5}{3!} $$ $$ = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} $$ $$ \binom{7}{3} = 35 $$