If at the end of certain meeting, everyone had shaken hands with everyone else, it was found that 45…
If at the end of certain meeting, everyone had shaken hands with everyone else, it was found that 45 handshakes were exchanged, then the number of members present at the meeting, are
10
15
20
21
Solution
Let ' $n$ ' be the number of members in the meeting
$\therefore \quad$ Total number of handshakes $={ }^{\mathrm{n}} \mathrm{C}_2$
$\therefore \quad{ }^{\mathrm{n}} \mathrm{C}_2=45$
$\begin{array}{ll} & \frac{\mathrm{n} !}{2 !(\mathrm{n}-2) !}=45 \\ & \frac{\mathrm{n}(\mathrm{n}-1)(\mathrm{n}-2) !}{2 \times(\mathrm{n}-2) !}=45 \\ & \mathrm{n}(\mathrm{n}-1)=90 \\ \therefore \quad & \mathrm{n}^2-\mathrm{n}-90=0 \\ & \mathrm{n}=10 \text { or } \mathrm{n}=-9 \text { (not possible) } \\ \therefore \quad & \mathrm{n}=10\end{array}$