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
Alan P.
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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