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In an examination, a question paper consists of 12 questions divided into two parts i.e., Part I and Part II containing 5 and 7 questions, respectively. A student is required to atempt 8 questions in all, selecting at least 3 from each part. In how many ways can a student select the questions?
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Student may select 8 questions according to following scheme.
$\begin{array}{lcc} & \text { I (5 questions) } & \text { II (7 questions) } \\ \text { (a) } & 3 & 5 \\ \text { (b) } & 4 & 4 \\ \text { (c) } & 5 & 3\end{array}$
If $P$ is the required number of ways, then
$\begin{aligned}
P &={ }^5 \mathrm{C}_3 \times{ }^7 \mathrm{C}_5+{ }^5 \mathrm{C}_4 \times{ }^7 \mathrm{C}_4+{ }^5 \mathrm{C}_5 \times{ }^7 \mathrm{C}_3 \\
&=\frac{5.4}{2.1} \times \frac{7.6}{2.1}+5 \times \frac{7 \times 6 \times 5}{3 \times 2 \times 1}+1 \times \frac{7 \times 6 \times 5}{3.2 .1} \\
&=10 \times 21+5 \times 35+35=210+175+35 \\
&=420 \text { ways }
\end{aligned}$
$\begin{array}{lcc} & \text { I (5 questions) } & \text { II (7 questions) } \\ \text { (a) } & 3 & 5 \\ \text { (b) } & 4 & 4 \\ \text { (c) } & 5 & 3\end{array}$
If $P$ is the required number of ways, then
$\begin{aligned}
P &={ }^5 \mathrm{C}_3 \times{ }^7 \mathrm{C}_5+{ }^5 \mathrm{C}_4 \times{ }^7 \mathrm{C}_4+{ }^5 \mathrm{C}_5 \times{ }^7 \mathrm{C}_3 \\
&=\frac{5.4}{2.1} \times \frac{7.6}{2.1}+5 \times \frac{7 \times 6 \times 5}{3 \times 2 \times 1}+1 \times \frac{7 \times 6 \times 5}{3.2 .1} \\
&=10 \times 21+5 \times 35+35=210+175+35 \\
&=420 \text { ways }
\end{aligned}$
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