From 8 names on a ballot, a committee of 3 will be elected to attend a political national convention. How many different committees are possible?

1 Answer
Apr 28, 2016

Answer:

#56# different committee are possible

Explanation:

This is a "how many ways can you choose #n# items from a pool of #m# items?" type of question.

The general form of the answer is "#m# Choose #n#";
written as:
#color(white)("XXX")mCn=(m!)/((m-n)!n!)#

In this case
#color(white)("XXX")8C3=(8!)/(5!*3!)#

#color(white)("XXXXXX")=(8xx7xx6xxcancel(5xx4xx3xx2xx1))/(cancel(5xx4xx3xx2xx1)*(3xx2))#

#color(white)("XXXXXX")=(8xx7xxcancel(6))/(cancel(3xx2))#

#color(white)("XXXXXX")=56#

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Another way to see this:

There are #8# people that could be our first choice;
after this there would be #7# people for our second choice;
and then #6# people for our third choice.
This would give us #8xx7xx6# ways of making first, second, and third choices.

But the same three people could form our 3 person committee in multiple ways. e.g. {Alice, Bob, Carol} and {Bob, Carol, Alice} should be considered as the same committee.

How many ways could 3 people be arranged among 3 sequential positions that should be treated as the same committee?
There are #3# possibilities for the first position; #2# for the second; and #1# for the third. So #3xx2xx1=6# arrangements should be considered the same committee.

Therefore the number of committees (without internal ordering) should be #(8xx7xx6)/(3xx2xx1) =56#