There are 6 boys and 3 girls in a class. An entertainment committee of 5 is to be selected such that their are 3 boys and 2 girls in a committee. In how many ways can the committee be selected? What are the number of ways if their is at least one girl?

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$$\begin{gathered} \mathrm{We} \mathrm{have} 6 \mathrm{boys} \mathrm{and} 3 \mathrm{girls}. \\ \mathrm{Number} \mathrm{of} \mathrm{ways} \mathrm{of} \mathrm{selecting} 3 \mathrm{boys}{=}^6{\mathrm{C}}_3 \\ \mathrm{Number} \mathrm{of} \mathrm{ways} \mathrm{of} \mathrm{selecting} 2 \mathrm{girls}{=}^3{\mathrm{C}}_2 \\ \Rightarrow \mathrm{Required} \mathrm{number} \mathrm{of} \mathrm{ways}{=}^6{\mathrm{C}}_3{\times }^3{\mathrm{C}}_2 \\ =\frac{6\times 5\times 4}{3\times 2\times 1}\times \frac{3\times 2}{2\times 1} \\ =20\times 3 \\ \mathbf{=}\mathbf{60}\left(\mathbf{Answer}\right) \\ \mathrm{As} \mathrm{there} \mathrm{should} \mathrm{be} \mathrm{atleast} \mathrm{one} \mathrm{girl}, \mathrm{hence} \mathrm{committee} \mathrm{can} \mathrm{be} \mathrm{as} \mathrm{follows}: \\ \left(4 \mathrm{boys} \mathrm{AND}\hspace{0.17em}1 \mathrm{girl}\right)\mathrm{OR}\left(3 \mathrm{boys} \mathrm{AND}\hspace{0.17em}2 \mathrm{girl}\right)\mathrm{OR}\left(2 \mathrm{boys} \mathrm{AND}\hspace{0.17em}3 \mathrm{girl}\right) \\ \Rightarrow \mathrm{Required} \mathrm{number} \mathrm{of} \mathrm{ways} \\ =\left({}^6{\mathrm{C}}_4{\times }^3{\mathrm{C}}_1\right)+\left({}^6{\mathrm{C}}_3{\times }^3{\mathrm{C}}_2\right)+\left({}^6{\mathrm{C}}_2{\times }^3{\mathrm{C}}_3\right) \\ =\left(\frac{6\times 5\times 4\times 3}{4\times 3\times 2\times 1}\times \frac{3}{1}\right)+\left(\frac{6\times 5\times 4}{3\times 2\times 1}\times \frac{3\times 2}{2\times 1}\right)+\left(\frac{6\times 5}{2\times 1}\times \frac{3\times 2\times 1}{3\times 2\times 1}\right) \\ =45+60+15 \\ \mathbf{=}\mathbf{120}\left(\mathbf{Answer}\right) \end{gathered}$$

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