KA-0163Selections of a few objects
4 points, difficulty 2 of 3Level 9–10about 105sA club has 8 members. In how many ways can a group of 3 be chosen to attend a conference, if the three places are all the same?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
The three places are identical.
2The strategy
There are 8 x 7 x 6 = 336 ways to pick three people in order. Each group of three has been counted several times.
3The full solution
Each group of 3 appears 3! = 6 times in that list, so the answer is 336 / 6 = 56.
Solution
The reliable way
Count the ordered picks first, then correct for the overcounting. Choosing three people in order gives 8 x 7 x 6 = 336 possibilities. But the places are identical, so a group like {Ana, Ben, Cara} has been counted once for each of the 3! = 6 orders those three could be named in. Dividing gives 336 / 6 = 56. The transferable idea: when order does not matter, count as if it did and then divide by the number of orderings.
The elegant way
This is 8 choose 3, which is (8 x 7 x 6) / (3 x 2 x 1) = 56.
Why this is on the test: Overcount-then-correct is the workhorse of every selection question, and dividing by the wrong number is the usual slip.