Correct Answer - Option 2 :

^{10}C

_{6}\(\rm (\frac{2}{5})^{6}(\frac{3}{5})^{4}\)
__Concept:__

Binomial distribution:

If a random variable X has binomial distribution as B (n, p) with n and p as parameters, then the probability of random variable is given as:

P( X = k) = nCk pk q(n - k) where q = p - 1, n is the number of observations, p is the probability of success & q is the probability of failure.

Note: The mean of a binomial distribution is np and variance is npq.

__Calculation:__

Given: The probability that a student passes in JEE mains exam is 3/5.

Here, we have to find the probability that out of 10 students, 6 students fails to pass the JEE mains exam

It implies that the probability of failure denoted by q is 3/5.

Let the probability of success is denoted by p.

⇒ p = 1 - q = 1 - 3/5 = 2/5.

Here, n = 10, k = 6, p = 2/5 and q = 3/5

As we know that, P( X = k) = nCk pk (1 - p)(n - k)

⇒ P( X = 6) = 10C_{6}\(\rm (\frac{2}{5})^{6}(\frac{3}{5})^{4}\)

Hence, the correct option is 2.