Solution: (a)

Number of combinations = \(\displaystyle ^{4}{{C}_{4}}{{\times }^{6}}{{C}_{1}}{{+}^{3}}{{C}_{3}}{{\times }^{4}}{{C}_{2}}=1\times 6+1\times 6=12\)

Solution: (c)

Number of combinations = Selecting 2 trainees out of 3 and selecting 3 research associates out of 6 =

\(\displaystyle ^{3}{{C}_{2}}{{\times }^{6}}{{C}_{3}}=3\times \frac{{6\times 5\times 4}}{{1\times 2\times 3}}=60\)

Solution: (d)

Total possible outcomes = Number of ways of picking 3 marbles out of 12 marbles = n(S)

\(\displaystyle ^{{12}}{{C}_{3}}=\frac{{12\times 11\times 10}}{{1\times 2\times 3}}=220\)

Favourable number of cases = n(E)

\(\displaystyle ^{3}{{C}_{3}}{{+}^{4}}{{C}_{3}}=1+4=5\)

Therefore, Required probability = \(\displaystyle \frac{{n(E)}}{{n(S)}}=\frac{5}{{220}}=\frac{1}{{44}}\)

Solution: (e)

Total possible outcomes = n(S) = \(\displaystyle ^{{12}}{{C}_{2}}=\frac{{12\times 11}}{{1\times 2}}=66\)

Favourable number of cases = n(E) = \(\displaystyle ^{4}{{C}_{2}}=\frac{{4\times 3}}{{1\times 2}}=6\)

Therefore, Required probability = \(\displaystyle \frac{{n(E)}}{{n(S)}}=\frac{{6}}{{66}}=\frac{{1}}{{11}}\)

Solution: (b)

Total possible outcomes = n(S) = \(\displaystyle ^{{12}}{{C}_{3}}=220\)

Favourable number of ways of picking 3 marbles (none is blue) out of 7 marbles = \(\displaystyle ^{7}{{C}_{3}}=\frac{{7\times 6\times 5}}{{1\times 2\times 3}}=35\)

Therefore, Required probability = \(\displaystyle (1-\frac{{35}}{{220}})=1-\frac{7}{{44}}=\frac{{37}}{{44}}\)