$\begin{array}{1 1}(A)\;180\\(B)\;190\\(C)\;200\\(D)\;210\end{array} $

- $C(n,r)=\large\frac{n!}{r!(n-r)!}$

Since every pair of the students gives us a handshake

$\therefore$ Total number of handshakes =Total number of pairs of students

$\Rightarrow $Number of ways of choosing two students out of 20 students

$\Rightarrow 20C_2=\large\frac{20!}{2!18!}$

$\Rightarrow \large\frac{20\times 19\times 18!}{2\times 18!}$

$\Rightarrow 190$

Hence (B) is the correct answer.

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