**Question:**

In the binomial expansion of $(a+b)^{n}$, the coefficients of the $4^{\text {th }}$ and $13^{\text {th }}$ terms are equal to each other. Find the value of $\mathrm{n}$.

**Solution:**

To find: the value of n with respect to the binomial expansion of (a + b)n where the coefficients of the 4th and 13th terms are equal to each other

Formula Used:

A general term, $T_{r+1}$ of binomial expansion $(x+y)^{n}$ is given by,

$T_{r+1}={ }^{n} C_{r} x^{n-r} y^{r}$ where

${ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}}=\frac{n !}{r !(n-r) !}$

Now, finding the $4^{\text {th }}$ term, we get

$\mathrm{T}_{4}={ }^{n} \mathrm{C}_{3} \times \mathrm{a}^{\mathrm{n}-3} \times(\mathrm{b})^{3}$

Thus, the coefficient of a $4^{\text {th }}$ term is ${ }^{n} C_{3}$

Now, finding the $13^{\text {th }}$ term, we get

$\mathrm{T}_{13}={ }^{n} \mathrm{C}_{12} \times \mathrm{a}^{\mathrm{n}-12} \times(\mathrm{b})^{12}$

Thus, coefficient of $4^{\text {th }}$ term is ${ }^{n} C_{12}$

As the coefficients are equal, we get

${ }^{n} C_{12}={ }^{n} C_{3}$

Also, ${ }^{n} C_{r}={ }^{n} C_{n-r}$

${ }^{n} \mathrm{C}_{n-12}={ }^{n} \mathrm{C}_{3}$

$n-12=3$

$\mathrm{n}=15$

Thus, value of n is 15