Let X have a binomial distribution B(n, p) such that the sum and...

Let $M =$Mean and $V =$Variance \left(M > V\right)$</p><p>So,$M + V = 24, MV = 128$</p><p>$⇒ M = 16$and$V = 8$</p><p>Now, mean$np = 16$</p><p>and variance$npq = 8 ⇒ q = \dfrac{1}{2}$</p><p>$∴ p = \dfrac{1}{2}, n = 32$</p><p>Now$P\left(X > n - 3\right) = \dfrac{1}{{2}^{n}}\left(Cn - 2 n + Cn - 1 n + Cn n\right)$</p><p>$∴ \dfrac{k}{{2}^{n}} = \dfrac{1}{{2}^{n}}\left(C30 32 + C31 32 + C32 32\right)$</p><p>$⇒ k = \dfrac{32 \times 31}{2} + 32 + 1$</p><p>$= 496 + 33 = 529