Question
A pizza shop offers a choice of 3 sizes (small, medium, and large) and 7 different toppings. Different pizzas can be created by changing the size and/or the choice of toppings. If Cody wants to order a pizza with exactly 2 different toppings, how many different pizzas can he create?
- A6
- B21
- C63
- D126
Show the answer
Correct answer: C
First, figure out how many different topping pairs are possible. Use 1, 2, 3, 4, 5, 6, 7 to represent the toppings and create a list of possible pairs:
1,2; 1,3; 1,4; 1,5; 1,6; 1,7
2,3; 2,4; 2,5; 2,6; 2,7
3,4; 3,5; 3,6; 3,7
4,5; 4,6; 4,7
5,6; 5,7
6,7
So there are 21 different topping combinations for one pizza. Since there are 3 pizza sizes, multiply the total number of combinations by 3 to get the total number of different pizzas Cody can create:
3 × 21 = 63
Question analysis
A pizza shop offers 3 sizes and 7 toppings, and Cody wants exactly 2 different toppings; students count how many different pizzas he could order. It tests counting combinations, where the order of the toppings does not matter, together with the multiplication principle. The difficulty is counting each topping pair exactly once and including every way the pizzas can differ.
- NYS standards
- 7.SP.8b sample spaces for compound events
- Grade level
- Grade 7 content, by NYS standard
- Compared with grade 7
- Above grade 7 Grade 7 students list outcomes for compound events, usually ordered pairs like a coin and a die. Choosing an unordered pair from seven items and combining it with sizes goes beyond that, and two traps are well placed.
- Estimated difficulty
- Hard: combines ideas, needs a plan, and has more than one tempting wrong answer.
- Thinking it demands
- Depth of knowledge 2, skills and concepts · mainly apply
- What makes it hard
- Combines ideas moderate
- Something you must already know minor
- Solution
- 3 steps
- Reading level
- Question and choices: about grade 8, 48 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
The question hides two separate counts behind one sentence, and both carry traps: order does not matter among toppings, while size multiplies everything. Students who list pairs systematically are safe on the first count but often forget the second, and students who reach for 7 × 6 double the pairs. The phrase "size and/or the choice of toppings" is the cue that sizes count too.
Combines ideas moderate Working it out
Two counting ideas are joined: choosing an unordered pair from 7 toppings, then multiplying by the 3 sizes, and either part can go wrong on its own. (Embretson, 1998; Daniel & Embretson, 2010)
Something you must already know minor Working it out
Students must see that a pair of toppings is the same pizza in either order, so the product 7 × 6 counts every pair twice. (Anderson & Krathwohl, 2001)
Strategy: List the topping pairs systematically (topping 1 with each later topping, then topping 2 with each later one, and so on), count them, then multiply by the number of sizes.
Traps
- Stops one step early: A choice is a number from partway through the solution.
- Overlapping groups: Things in two groups at once are counted twice, or not subtracted.
The wrong answers, classified
A Numbers taken from the question (Hegarty, Mayer & Monk, 1995)
Multiplies the 3 sizes by the 2 toppings, using the number of toppings chosen instead of the number of ways to choose them.
B Stops one step early (King et al., 2004; Gierl et al., 2017)
Counts the 21 topping pairs correctly but stops without multiplying by the 3 sizes each pair can come in.
D Overlapping groups (King et al., 2004)
Counts ordered pairs, 7 × 6 = 42 for each size, so each pair of toppings is counted twice, then multiplies by 3 to get 126.
Worked solution, step by step
Formulas used
- Counting principle
- a is the number of ways to make the first choice. b is the number of ways to make the second choice. Each first choice can go with every second choice, so you multiply to count them all.
- Number of pairs from a group
- n is how many things you can pick from. n times (n − 1) counts a first pick times a second pick. But that counts each pair twice, once in each order. Dividing by 2 fixes that. Use it when order does not matter.
Sort out the question
There are 3 sizes and 7 toppings. Cody picks one size and exactly 2 different toppings. Order does not matter for the toppings. Ham and olives is the same pizza as olives and ham. You need the number of different pizzas.
List the topping pairs
Number the toppings 1 to 7. Pair topping 1 with each later topping. Then pair topping 2 with each later one, and so on.
- 1 with 2, 3, 4, 5, 6 or 7: 6 pairs
- 2 with 3 to 7: 5 pairs
- 3 with 4 to 7: 4 pairs
- Then 3, 2 and 1 more pairs
There are 21 topping pairs.
Check the pairs with the formula
Use the pairs formula with 7 toppings.
This matches the list.
Watch out: counts each pair twice. Forgetting to divide by 2 gives , choice D.
Multiply by the three sizes
Each topping pair can come in any of the 3 sizes. Use the counting principle.
Watch out: stopping at 21 forgets the sizes. That gives choice B.
Give the answer
There are 21 topping pairs and 3 sizes for each. So the answer is C, 63 different pizzas.
Check your answer
Test the pairs idea on a smaller case. With 3 toppings, the pairs are 1 and 2, 1 and 3, and 2 and 3. That is 3 pairs. The formula gives . It works, so 21 pairs and 63 pizzas are right.
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