A sports team of 11 students is to be constituted, choosing at least 5
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 team be constituted? 

Solution:

There are 20 students in each classes and there is need of at least 5 students in each class to form a team of team of 11.

Now

There are two ways in which the selection can be possible

1. Selecting 5 from $\mathrm{XI}$ and 6 from $\mathrm{XII}$

2. Selecting 6 from $\mathrm{XI}$ and 5 from $\mathrm{XII}$

Now, considering first case,

No. of ways in selection of 5 students from 20 in class $\mathrm{XI}={ }^{20} \mathrm{C}_{5}$

No. of ways in selection of 6 students from 20 in class $X I I={ }^{20} C_{6}$

By multiplication principle total no. of ways in first case is

$={ }^{20} \mathrm{C}_{5} \times{ }^{20} \mathrm{C}_{6}$

Now, considering second case,

No. of ways in selection of 6 students from 20 in class $\mathrm{XI}={ }^{20} \mathrm{C}_{6}$

No. of ways in selection of 5 students from 20 in class $\mathrm{XII}={ }^{20} \mathrm{C}_{5}$

By multiplication principle total no. of ways in second case is

$={ }^{20} \mathrm{C}_{6} \times{ }^{20} \mathrm{C}_{5}$

Now the total no. of ways will be the addition of both the cases

$={ }^{20} C_{5} \times{ }^{20} C_{6}+{ }^{20} C_{6} \times{ }^{20} C_{5}$

$=2^{\times{ }^{20} C_{6}} \times{ }^{20} C_{5}$

Thus these are the ways by which A sports team of 11 students is to be constituted 

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