Three numbers are selected at random (without replacement) from the first six positive integers. Let X denote the largest of the three numbers obtained. Find the probability distribution of X. Also, find the mean and variance of the distribution.

First six positive integers are 1, 2, 3, 4, 5, 6.
Therefore, Three numbers are selected at random without replacement so, total no. of ways 6C3 = 20.
Let, x denote the larger of three numbers.
So, x can take values 3, 4, 5, 6.
p(x=3) = 1/20
p(x=4) = 3/20
p(x=5) = 6/20
p(x=6) = 10/20
Refer the tabular column:

mean = Σxp(x) = 3/20 + 12/20 + 30/20 + 60/20 = 105/20 = 5.25
Variance = Σx²p(x) - (Σxp(x))² = (567/20) - (105/20)² = 28.35 - 27.5625 = 0.7875