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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 $15$ $15$ guests

Explanation:

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

Therefore with $n$ $n$ guests the number of handshakes will be $\frac{n \times (n - 1)}{2}$ $(nxx(n-1))/2$

We are told
$XXX \frac{n \times (n - 1)}{2} = 105$ $color(white)("XXX")(nxx(n-1))/2=105$

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

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

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

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

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