Animated Solution for Mathematics - Probability: Let $X$ be a random variable having binomial distribution $B(7,p)$. If $P(X=3)=5P(X=4)$, then the sum of the mean and the variance of $X$ is:

Welcome, my dear student, to the fascinating world of probability. Today, we are going to explore the Binomial Distribution, a cornerstone of statistical mechanics and decision theory.

Imagine a random variable $X$ that follows a binomial distribution $B(7,p)$. This means we are conducting $n=7$ independent trials, where each trial has a probability of success $p$ and a probability of failure $q=1-p$.

The probability of getting exactly $k$ successes is given by the elegant formula:

This formula is our recipe for success.

The problem presents us with a beautiful condition: $P(X=3)=5P(X=4)$. This is not just an equation; it is a constraint that defines the very nature of our distribution.

For $k=3$, we have $P(X=3)={}^7{C}_3{p}^3{q}^4$. For $k=4$, we have $P(X=4)={}^7{C}_4{p}^4{q}^3$.

Now, we set them equal according to the condition:

We invoke the symmetry property of combinations: ${}^n{C}_r={}^n{C}_{n-r}$. Since $7-3=4$, we know that ${}^7{C}_3$ is exactly equal to ${}^7{C}_4$.

They cancel out! We are left with $p^3{q}^4=5p^4{q}^3$. Dividing both sides by $p^3{q}^3$, we get the remarkably simple relation:

Since we know $q=1-p$, we can substitute this to get $1-p=5p$, which leads us directly to $6p=1$, or $p=\frac{1}{6}$. Consequently, $q=\frac{5}{6}$.

Now that we have unlocked the values of $p$ and $q$, the rest is a victory lap. The problem asks for the sum of the mean and the variance of $X$.

For a binomial distribution, the mean $\mu$ is $np$, and the variance ${\sigma }^2$ is $npq$. Calculating the mean, we get:

Calculating the variance, we get:

Finally, we sum them:

And there you have it! Through the power of symmetry and algebraic simplification, we have arrived at the final answer of $\frac{77}{36}$. Remember, in JEE, the math is never just about calculation; it is about finding the elegant path through the complexity.