KA-0074Selections of a few objects
4 points, difficulty 2 of 3Level 5–6about 75sTwo students are chosen from a group of six to represent the class. How many different pairs are possible?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
Order does not matter here.
2The strategy
Count the ordered choices first: six options for the first student, five for the second. Then ask how many times each unwanted pair got counted, and correct for it.
3The full solution
6 x 5 = 30 ordered choices, but each pair appears twice, so 30 divided by 2 = 15.
Solution
The reliable way
Choosing in order gives 6 x 5 = 30. But representing the class is not an ordered job — picking Ana then Bo is the same pair as Bo then Ana — so every pair has been counted twice. Halving gives 15. The transferable idea: when order does not matter, count the ordered version and divide by the number of orders.
The elegant way
Each of the 6 students pairs with 5 others, giving 30 ends of pairs, and each pair has two ends: 15.
Why this is on the test: Selections and arrangements look identical on the page; only the word order tells them apart, and forgetting to halve is the standard slip.