A sports team of 11 students is to be constituted, choosing at least 5 from class XI and at least 5 from class XII.
Question:

A sports team of 11 students is to be constituted, choosing at least 5 from class XI and at least 5 from class XII. If there are 20 students in each of these classes, in how many ways can the teams be constituted?

Solution:

A sports team of 11 students is to be constituted, choosing at least 5 students of class XI and at least 5 from class XII.

Required number of ways $={ }^{20} C_{5} \times{ }^{20} C_{6}+{ }^{20} C_{6} \times{ }^{20} C_{5}=2 \times{ }^{20} C_{5} \times{ }^{20} C_{6}=2\left(20_{C_{6}} \times 20_{C_{5}}\right)$