CMAT 2019 Solved Question Paper

Probability $=\frac{\mathrm{Expected}\mathrm{number}\mathrm{of}\mathrm{outcomes}}{\mathrm{Total}\mathrm{number}\mathrm{of}\mathrm{outcomes}}$.
Total number of outcomes = The number of ways of selecting 3 professors at random from a total of 10 professors.
= ${}_{}^{10}{\mathit{C}}_3^{}$ = 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 ${=}^4{\mathit{C}}_1{\times }^6{\mathit{C}}_2$.
Case(2): Selecting 2 junior professor out of 4 and Selecting 1 senior professor out of 6 ${=}^4{\mathit{C}}_2{\times }^6{\mathit{C}}_1$.
Case (3): Selecting 3 junior professor out of 4 ${=}^4{\mathit{C}}_3$.
So Expected number of outcomes = ${}_{}^4{\mathit{C}}_1^{}{\times }^6{\mathit{C}}_2$ ${+}^4{\mathit{C}}_2{\times }^6{\mathit{C}}_1{+}^4{\mathit{C}}_3$ $=4\times 15\times 6\times 6+4$ = 100 ways.
Probability $=\frac{100}{120}=\frac{5}{6}$