Selective Shaking

Probability Level 2

At a recent party, everyone shook hands exactly once with everyone else. Halfway through the party Magda arrived and shook hands only with the people she likes.

In all there were 201 handshakes.

How many people does Magda like?

The answer is 11.

1 solution

Nicholas James
Mar 3, 2017

If everyone in a group of $n$ people shakes hands with everyone else, there are $\binom{n}{2}=\frac{n(n-1)}{2}$ handshakes.

People at party	Handshakes
$19$	$\binom{19}{2}=171$
$20$	$\binom{20}{2}=190$
$21$	$\binom{21}{2}=210$

We can see that $19$ is too few, as Magda would have to shake hands with more people than are present at the party. Similarly, $21$ is too many, as there would have already been $210$ handshakes before she arrived.

Therefore, there are $20$ people at the party, and Magda likes $201-190=\boxed{11}$ of them.

Selective Shaking

The answer is 11.

1 solution

0 pending reports