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At a party each guest shook hands with every other guest exactly once. There were a total of 105 handshakes. How many guest were there?

1 Answer

Jan 22, 2016

There are $15$ guests

Explanation:

If there are $n$ guests then each guest shakes hands with $(n-1)$ other guests. Note, however, that each time a handshake occurs it counts as $2$ handshakes (one for each person involved).

Therefore with $n$ guests the number of handshakes will be $(nxx(n-1))/2$

We are told
$color(white)("XXX")(nxx(n-1))/2=105$

Therefore
$color(white)("XXX")n^2-n=210$

$color(white)("XXX")n^2-n-210=0$

$color(white)("XXX")(n+14)(n-15)=0$

Therefore
$color(white)("XXX")n=-14$ or $n=15$
And since the number of guest can not be negative:
$color(white)("XXX")n=15$

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