Let a random variable X has a binomial distribution with mean 8 and variance 4. If  $P(X\le 2)=\frac{k}{2^{16}}$ , then the value of k is equal to

Let n be the number of time the experiment is done, p is the probability of success and q is the probability of failure
$\therefore np=8\left(mean\right),npq=4\left(\text{variance}\right)$
$\Rightarrow q=\frac{1}{2}\Rightarrow p=\frac{1}{2}$
$\Rightarrow n=16$
$P(X\le 2)=p(X=0)+p(X=1)+p(X=2)$
$={}^{16}{C}_0{p}^0{q}^{16}+{}^{16}{C}_1{p}^1{q}^{15}+{}^{16}{C}_2{p}^2{q}^{14}$
$={\left(\frac{1}{2}\right)}^{16}\left(137\right)=\frac{137}{2^{16}}=\frac{k}{2^{16}}$
$\therefore k=137$