A management institute has 6 senior professors and 4 junior professors, 3 professors are selected at random for a government project. The probability that at least one of the junior professors would get selected is :

Probability = Expected number of outcomes/ Total number of outcomes.

Total number of outcomes =  The number of ways of selecting 3 professors at random from a total of 10 professors.

= 10C3 = 120 ways.

Expected number of outcomes = Selecting 3 professors so that at least one of the junior professor is selected.

The possible cases are,

Case(1): Selecting 1 junior professor out of 4 and Selecting 2 senior professor out of 6 = 4C1 * 6C2.

Case(2): Selecting 2 junior professor out of 4 and Selecting 1 senior professor out of 6 = 4C2 * 6C1.

Case (3): Selecting 3 junior professor out of 4  = 4C3.

So Expected number of outcomes = 4C1 * 6C2 + 4C2 * 6C1 + 4C3 = $$4\times 15 + 6\times 6 + 4$$ = 100 ways.

Probability = $$\frac{100}{120} = \frac{5}{6}$$