Also in 2026 Practice Test A Math, question 29; 2025 Practice Test A Math, question 34.
Question
There are 6 different cookies on a plate. Aiden will choose 2 of these cookies to pack in his lunch. How many different pairs of 2 cookies can he choose from the 6?
- A12
- B15
- C30
- D36
Show the answer
Correct answer: B
Create a list of the possible pairs. Let the cookies be named A, B, C, D, E, and F.
AB, AC, AD, AE, AF
BC, BD, BE, BF
CD, CE, CF
DE, DF
EF
There are a total of 15 possible pairs of cookies that Aiden can choose.
Question analysis
This counting question asks how many different pairs of cookies Aiden can choose from six different cookies on a plate. Students must count combinations, where taking the same two cookies in a different order is not a new pair. The challenge is that the order of picking does not matter, so each pair must be counted once.
- NYS standards
- 7.SP.8b sample spaces for compound events
- Grade level
- Grade 7 content, by NYS standard
- Compared with grade 7
- At grade 7 Grade 7 students list the outcomes of compound events with organized lists and tree diagrams. Counting pairs is close to that work, but students must notice that the order of choosing does not matter.
- Estimated difficulty
- Moderate: several steps, or a close reading with one strong distractor.
- Thinking it demands
- Depth of knowledge 2, skills and concepts · mainly apply
- What makes it hard
- Much to keep track of moderate
- Solution
- 2 steps
- Reading level
- Question and choices: about grade 4, 29 words.
- Students, last 30 days
- Not enough answers yet to show (it takes 10).
Why it is hard, traps and wrong answers (shows the answer)
Why it is hard
Students who know the multiplication principle are the most at risk, because it counts each pair twice. The question rewards noticing that a pair has no order, either by listing systematically or by halving 6 × 5. The list of 15 pairs takes under a minute, so there is no time pressure to justify a shortcut.
Much to keep track of moderate Working it out
Listing all pairs by hand means tracking which have been written so none is repeated or skipped, which an organized list such as AB to AF handles. (Sweller, 1988; Carpenter, Just & Shell, 1990)
Strategy: Name the cookies with letters and list the pairs systematically: the first cookie with every later one, then the second with every later one, and so on.
Traps
- Overlapping groups: Things in two groups at once are counted twice, or not subtracted.
- Without replacement: In a second pick there is one fewer item, which changes the probability.
The wrong answers, classified
A Numbers taken from the question (Hegarty, Mayer & Monk, 1995)
Multiplies the 6 cookies by the 2 being chosen, a product of the stem's numbers that does not count any pairs.
C Overlapping groups (King et al., 2004)
Counts ordered picks, 6 choices then 5, so every pair such as AB and BA is counted twice, giving 30 instead of half that.
D Without replacement (Fischbein & Schnarch, 1997)
Uses 6 × 6, as if the first cookie went back on the plate, which counts order and also allows the same cookie twice.
Worked solution, step by step
Formulas used
- Number of pairs
- n is how many things you choose from. Each of the n things can pair with the n − 1 others. That counts every pair twice, once from each side, so you divide by 2.
Sort out the question
There are 6 different cookies. Aiden picks 2 of them. A pair is just two cookies together.
Order does not matter. Picking cookie 1 then cookie 2 gives the same pair as cookie 2 then cookie 1. You need to count the different pairs.
List the pairs in order
Number the cookies 1 to 6. Pair cookie 1 with every cookie after it. Then pair cookie 2 with every cookie after it, and so on. Only pairing with later numbers stops you from listing a pair twice.
Start with Pairs How many 1 1-2, 1-3, 1-4, 1-5, 1-6 5 2 2-3, 2-4, 2-5, 2-6 4 3 3-4, 3-5, 3-6 3 4 4-5, 4-6 2 5 5-6 1 Add up the pairs
Each row has one fewer pair than the row above it. That is because one fewer cookie comes after it. Add the last column of the table:
There are 15 different pairs.
Check with the pairs formula
Put n = 6 into the formula:
Watch out: 6 × 5 = 30 counts each pair twice, once for each order. Stopping there gives choice C. And 6 × 6 = 36, choice D, lets Aiden pick the same cookie twice. So the answer is B, 15.
Check your answer
Think of handshakes. Each of 6 people shakes hands with the 5 others. That looks like 30 handshakes, but each handshake has two people in it, so it was counted twice. 30 ÷ 2 = 15, which matches the list.
More questions like this
The original DOE question
