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A sports team of 11 students is to be constituted, choosing atleast 5 from class XI and atleast 5 from class XII. If there are 20 students in each of these classes, in how many ways can the team be constituted?
Solution:
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Verified Answer
Total students in each class $=20$
And we have to select atleast 5 students from each class Hence, selection of sports team of 11 students from each class is given as:
$$
\begin{array}{lll}
\hline \text { Class XI } & 5 & 6 \\
\text { Class XII } & 6 & 5 \\
\hline
\end{array}
$$
$\therefore$ Total number of ways of selecting a team of 11 players
$$
\begin{gathered}
=\left({ }^{20} C_5 \times{ }^{20} C_6\right)+\left({ }^{20} C_6 \times{ }^{20} C_5\right) \\
=2 \times{ }^{20} C_5 \times{ }^{20} C_6
\end{gathered}
$$
And we have to select atleast 5 students from each class Hence, selection of sports team of 11 students from each class is given as:
$$
\begin{array}{lll}
\hline \text { Class XI } & 5 & 6 \\
\text { Class XII } & 6 & 5 \\
\hline
\end{array}
$$
$\therefore$ Total number of ways of selecting a team of 11 players
$$
\begin{gathered}
=\left({ }^{20} C_5 \times{ }^{20} C_6\right)+\left({ }^{20} C_6 \times{ }^{20} C_5\right) \\
=2 \times{ }^{20} C_5 \times{ }^{20} C_6
\end{gathered}
$$
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