worksheet a2 fundamental counting principle Doc Template | pdfFiller - Free Printable

Let's solve each problem on Worksheet A2: Fundamental Counting Principle, Factorials, Permutations Intro step by step.

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1. A restaurant offers four types of pizza, two types of crust, and eight toppings. How many possible combinations of pizza with one topping are there?

We are choosing:
- 1 type of pizza (4 choices)
- 1 type of crust (2 choices)
- 1 topping (8 choices)

Using the Fundamental Counting Principle, multiply the number of choices:

$$
4 \times 2 \times 8 = 64
$$

✔ Answer: 64 combinations

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2. How many ways can 5 paintings be line up on a wall?

This is a permutation problem — we are arranging all 5 distinct paintings in order.

Number of permutations of 5 items:

$$
5! = 5 \times 4 \times 3 \times 2 \times 1 = 120
$$

✔ Answer: 120 ways

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3. Beth has 4 shirts, 3 pairs of pants, and 2 pairs of shoes that all coordinate. How many outfits can she put together?

Each outfit consists of:
- 1 shirt (4 choices)
- 1 pair of pants (3 choices)
- 1 pair of shoes (2 choices)

Use the Fundamental Counting Principle:

$$
4 \times 3 \times 2 = 24
$$

✔ Answer: 24 outfits

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4. Grace loves to eat salad! How many salads can she put together if she can pick out one type of lettuce (out of 2 choices), one vegetable (out of 3 choices), and one dressing (out of 7 choices)?

Choices:
- Lettuce: 2
- Vegetable: 3
- Dressing: 7

$$
2 \times 3 \times 7 = 42
$$

✔ Answer: 42 different salads

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5. An ice cream parlor has 3 flavors followed by 2 numbers.

#### a) If the same letter or number can be repeated, how many can be made?

Assuming "letters" means flavors (say A, B, C), and "numbers" are digits (like 1–9 or 0–9). But since it says "followed by 2 numbers", let's assume digits from 0–9.

So:
- 3 flavor choices
- First number: 10 choices (0–9)
- Second number: 10 choices (0–9)

$$
3 \times 10 \times 10 = 300
$$

✔ Answer: 300 possible combinations

#### b) If the same letter/number CANNOT be repeated, how many can be made?

Now, letters and numbers must be unique.

But note: "same letter or number cannot be repeated" — this likely means:
- The two numbers must be different
- The flavor is still just one choice

So:
- Flavor: 3 choices
- First number: 10 choices
- Second number: 9 choices (since it can't repeat the first)

$$
3 \times 10 \times 9 = 270
$$

✔ Answer: 270 combinations

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6. How many 3-digit numbers can be formed using 0 – 9?

We are forming 3-digit numbers (e.g., 123, 456, etc.), so the first digit cannot be 0.

Break it down:
- Hundreds place: cannot be 0 → 9 choices (1–9)
- Tens place: any digit → 10 choices (0–9)
- Units place: any digit → 10 choices (0–9)

$$
9 \times 10 \times 10 = 900
$$

✔ Answer: 900 three-digit numbers

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7. How many 5-digit numbers can be formed if each one uses all the digits 1, 2, 3, 4, 5 without repetition?

We are using all five digits exactly once, so it’s a permutation of 5 distinct digits.

$$
5! = 120
$$

✔ Answer: 120 five-digit numbers

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8. In how many ways can 6 bicycles be parked in a row?

This is like arranging 6 distinct objects in a line.

$$
6! = 720
$$

✔ Answer: 720 ways

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9. Evaluate (show all your work):

#### a) $ 6! $

$$
6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720
$$

✔ Answer: 720

#### b) $ 9! $

$$
9! = 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1
$$

Break it down:
- $ 9 \times 8 = 72 $
- $ 72 \times 7 = 504 $
- $ 504 \times 6 = 3024 $
- $ 3024 \times 5 = 15120 $
- $ 15120 \times 4 = 60480 $
- $ 60480 \times 3 = 181440 $
- $ 181440 \times 2 = 362880 $
- $ 362880 \times 1 = 362880 $

✔ Answer: 362,880

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✔ Final Answers Summary:

| Problem | Answer |
|--------|--------|
| 1 | 64 |
| 2 | 120 |
| 3 | 24 |
| 4 | 42 |
| 5a | 300 |
| 5b | 270 |
| 6 | 900 |
| 7 | 120 |
| 8 | 720 |
| 9a | 720 |
| 9b | 362,880 |

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