From 11 names on a ballot, a committee of 6 will be elected to attend a political national convention. How many different committees are possible?
Answers
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
8
×
7
×
6
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
3
×
2
×
1
=
6
arrangements should be considered the same committee.
Therefore the number of committees (without internal ordering) should be
8
×
7
×
6
3
×
2
×
1
=
5