A fair coin is tossed $n$ times. If the probability that heads occurs $6$ times is equal to the probability that heads occurs $8$ times, then value of $n$ is 

Let $X$ be the number of times heads occurs. Then $X$ follows a binomial distribution with parameter $n$ and $p=1/2.$ We are given that $P(X = 6) = P(X = 8).$ 

$\begin{array}{l}\Rightarrow { }^n{C}_6{\left(\frac{1}{2}\right)}^6{\left(\frac{1}{2}\right)}^{n-6}{=}^n{C}_8{\left(\frac{1}{2}\right)}^8{\left(\frac{1}{2}\right)}^{n-8}\\ \Rightarrow { }^n{C}_6{\left(\frac{1}{2}\right)}^n{=}^n{C}_8{\left(\frac{1}{2}\right)}^n\\ \Rightarrow { }^n{C}_6{=}^n{C}_8{=}^n{C}_{n-8}\end{array}$

$\Rightarrow 6=n-8$ 

$\Rightarrow n=14$