Francesca is picking out material for her new quilt.  At the fabric store, there are 17 solids, 7 striped prints, and 9 floral prints that she can choose from.  If she needs 2 solids, 3 floral prints, and 4 striped fabrics for her quilt, how many different ways can she choose the materials?

To find how many different ways Francesca can choose the materials for her quilt, we use combinations, since the order in which she selects the fabrics doesn't matter. The general formula for combinations is:

\[
C(n, k) = \frac{n!}{k!(n-k)!}
\]
where \( n \) is the total number of items to choose from, and \( k \) is the number of items being chosen.

Francesca needs:
- 2 solids from 17 available solids
- 3 floral prints from 9 available floral prints
- 4 striped fabrics from 7 available striped prints

We can calculate the number of ways to choose each type of fabric separately and then multiply them together to get the total number of combinations.

### 1. Choosing 2 solids from 17:
\[
C(17, 2) = \frac{17!}{2!(17-2)!} = \frac{17 \times 16}{2 \times 1} = 136
\]

### 2. Choosing 3 floral prints from 9:
\[
C(9, 3) = \frac{9!}{3!(9-3)!} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84
\]

### 3. Choosing 4 striped fabrics from 7:
\[
C(7, 4) = \frac{7!}{4!(7-4)!} = \frac{7 \times 6 \times 5 \times 4}{4 \times 3 \times 2 \times 1} = 35
\]

### Total number of combinations:
The total number of ways to choose the materials is the product of the combinations for each type of fabric:
\[
\text{Total combinations} = C(17, 2) \times C(9, 3) \times C(7, 4) = 136 \times 84 \times 35 = 399840
\]

Thus, there are **399,840** different ways Francesca can choose the materials for her quilt.

Would you like a breakdown of any specific part, or have any questions?

Here are 5 related questions:
1. How would the result change if she needed 3 solids instead of 2?
2. What if she only had 5 floral prints to choose from?
3. Can you calculate how many ways she can choose if she doesn't use floral prints at all?
4. What if she could use only one of each type of fabric?
5. How would the combinations change if she used just 1 floral print?

**Tip**: Combinations are used when the order of selection doesn't matter; permutations are used when order does matter.

Francesca is picking out material for her new quilt. At the fabric store, there are 17 solids, 7 striped prints, and 9 floral prints that she can choose from. If she needs 2 solids, 3 floral prints, and 4 striped fabrics for her quilt, how many different ways can she choose the materials?

Discover how to solve a combinatorics problem involving fabric choices for a quilt. Francesca needs to choose 2 solids from 17, 3 floral prints from 9, and 4 striped fabrics from 7. Learn how to use combinations to find the total number of ways she can make her selections. Ideal for students in Grades 9-12.

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