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# A sports team of 11 students is to be constituted,choosing at least 5 from class XI and 6 from XII.If there are 20 students in each of these classes,in how many ways can the team be constituted?

$\begin{array}{1 1}(A)\;2[20C_5.20C_5]\\(B)\;[20C_5.20C_5]\\(C)\;20C_5\\(D)\;20C_6\end{array}$

Toolbox:
• $C(n,r)=\large\frac{n!}{r!(n-r)!}$
A team of 11 students can be contituted in the following two ways:-
(i) 5 students from class XI and 6 from XII
(ii) 6 students from class XI and 5 from XII
$\therefore$ The required no of ways :-
$\Rightarrow C(20,5).C(20,6)+C(20,6).C(20,5)$
$\Rightarrow \large\frac{20\times 19\times 18\times 17\times 16\times 15!}{5\times 4\times 3\times 2\times 1\times 15!}\times \frac{20\times 19\times 18\times 17\times 16\times 15\times 15\times 14!}{6\times 5\times 4\times 3\times 2\times 14!}$
$\Rightarrow 2[19\times 3\times 17\times 16\times 19\times 17\times 8\times 15]$
$\Rightarrow 19^2\times 17^2\times 16\times 16\times 45$
$\Rightarrow 16^2.17^2.19^2.45$
Hence (A) is the correct answer.