Let X be a random variable having binomial distribution B(7, p) ....

Let probability of success is $p and failure is$q$<br />Given$n = 7$,</p><p>also ,$P\left(X = 3\right) = 5P\left(X = 4\right) ⇒ C3 7 {p}^{3}{q}^{4} = 5 \cdot C4 7 {p}^{4}{q}^{3}$</p><p>$⇒ q = 5p . . . \left(i\right)$</p><p>We know that$p + q = 1 ⇒ 6p = 1$(from \left(i\right)$)

$∴ p = \dfrac{1}{6}$ and $q = \dfrac{5}{6}$

$∴$ Mean $= np = 7 \cdot \dfrac{1}{6} = \dfrac{7}{6}$

and Variance $= npq = 7 \cdot \dfrac{1}{6} \cdot \dfrac{5}{6} = \dfrac{35}{36}$

$∴$ Sum of mean and variance $= \dfrac{7}{6} + \dfrac{35}{36} = \dfrac{77}{36}$