Let's solve each problem step by step and match it to the correct answer from the list (a–h). We'll use principles of
permutations (order matters) and
combinations (order doesn't matter), as appropriate.
---
1. How many different ways can you have 1st, 2nd, 3rd, and 4th in a race with 10 runners?
This is a
permutation problem because order matters (1st, 2nd, etc.).
We are choosing 4 positions out of 10, where order matters:
$$
P(10, 4) = 10 \times 9 \times 8 \times 7 = 5040
$$
✔ Answer: d. 5040
---
2. How many different combinations of management can there be to fill the positions of manager, secretary, and advisor of a company knowing that there are 12 eligible candidates?
Positions are distinct (manager, secretary, advisor), so
order matters → permutation.
We're assigning 3 distinct roles from 12 people:
$$
P(12, 3) = 12 \times 11 \times 10 = 1320
$$
✔ Answer: h. 1320
---
3. How many different words can we make using the letters S, H, A, and L? This assumes every combination possible is a word.
We are arranging all 4 distinct letters → number of permutations of 4 items:
$$
4! = 4 \times 3 \times 2 \times 1 = 24
$$
✔ Answer: g. 24
---
4. Robin has 5 different pairs of shoes that match with 6 different pairs of socks. How many shoes-and-socks combinations can he make if he selects one pair of shoes and one pair of socks?
This is a
multiplication principle — independent choices:
$$
5 \text{ shoes} \times 6 \text{ socks} = 30 \text{ combinations}
$$
✔ Answer: b. 30
---
5. Ginny has 6 water colors (red, green, blue, yellow, black and orange). In how many different orders can the different water color appear?
Arranging all 6 colors → permutations of 6 items:
$$
6! = 720
$$
✔ Answer: f. 720
---
6. How many different sandwiches can be made with 6 packets of bread, 4 packets of cheese, and 3 packets of potato if one choice is selected from each category?
One choice from each category → multiply options:
$$
6 \times 4 \times 3 = 72
$$
✔ Answer: c. 72
---
7. How many 2-digit numbers can you make using the digits 1, 2, 3, and 4 without repeating the digits?
We need 2-digit numbers with no repetition.
- First digit (tens place): 4 choices
- Second digit (units place): 3 remaining choices
$$
4 \times 3 = 12
$$
✔ Answer: a. 12
---
8. Eva has 8 glasses of water. In how many different orders can the glasses of water be arranged?
Assuming all glasses are
distinct, even if they contain the same liquid, the arrangement of 8 distinct objects is:
$$
8! = 40320
$$
✔ Answer: e. 40320
---
✔ Final Matching Answers:
| Problem | Answer Letter |
|--------|---------------|
| 1 | d |
| 2 | h |
| 3 | g |
| 4 | b |
| 5 | f |
| 6 | c |
| 7 | a |
| 8 | e |
---
✔ Completed Worksheet:
```
______ 1. d
______ 2. h
______ 3. g
______ 4. b
______ 5. f
______ 6. c
______ 7. a
______ 8. e
```
Let me know if you'd like this formatted for printing or want explanations in simpler terms!
Parent Tip: Review the logic above to help your child master the concept of permutation and combination worksheet.
