https://file.aiquickdraw.com/mathbot/file/1730048165411r760ide5.jpg

Let's break down the problem.

Initially, Susan invited 13 friends and created gift hampers that included:
- One $3 gift certificate
- One $4 gift certificate
- One $5 gift certificate

Since each hamper contained these three certificates, each had a total value of:
\[
3 + 4 + 5 = 12 \text{ dollars}
\]

However, one friend did not turn up, so she reconfigured the 13 hampers for the remaining 12 friends. Each new gift hamper is now supposed to be worth $13. 

### Step 1: Total Value of Original Certificates
Since each original hamper was worth $12, the total value of all 13 hampers was:
\[
13 \times 12 = 156 \text{ dollars}
\]

### Step 2: Total Value of New Configuration
Susan reconfigured the gift hampers for 12 friends, each receiving a $13 gift certificate. So, the total value needed in this new configuration is:
\[
12 \times 13 = 156 \text{ dollars}
\]

Since the total value remains the same, she only had to rearrange the certificates.

### Step 3: Determine the Composition of Each $13 Hamper
To create a $13 value using $3, $4, and $5 certificates, let’s explore combinations that add up to $13. We need to determine if a $5 certificate was used in each configuration.

1. **Combination without $5 certificate**:
   - Using two $3 certificates and one $4 certificate:
     \[
     3 + 3 + 4 = 10 \text{ dollars (not enough)}
     \]

2. **Combination with $5 certificate**:
   - Using one $5 certificate, one $4 certificate, and one $3 certificate:
     \[
     5 + 4 + 3 = 13 \text{ dollars}
     \]

Since we need exactly $13 in each hamper, every reconfigured gift hamper includes one $5 certificate to reach the required value.

### Conclusion
Since each hamper now must include a $5 certificate to reach the value of $13, there are **zero gift hampers** without a $5 certificate in the new configuration.

---

Would you like further details on any part of this solution?

Here are some related questions for practice:
1. What if the total required per gift hamper was $14 instead of $13?
2. How many $3 certificates would Susan need if each gift hamper required exactly two $3 certificates?
3. If a $6 certificate was added to the available options, how could that change the reconfiguration?
4. How would the solution change if two friends didn't show up?
5. Can you create a different arrangement of certificates totaling $13?

**Tip**: When tackling combinatorial problems, start by determining possible combinations that satisfy the conditions. This approach often helps reveal constraints in the problem.

Susan invited 13 of her friends for her birthday party and created return gift hampers comprising one each of $3, $4, and $5 gift certificates. One of her friends did not turn up and Susan decided to rework her gift hampers such that each of the 12 friends who turned up got $13 worth gift certificates. How many gift hampers did not contain $5 gift certificates in the new configuration?

Learn how to solve a combinatorial math problem involving gift certificates reconfigured for a birthday party. This problem involves creating $13 gift hampers from $3, $4, and $5 certificates after one friend didn’t turn up. Understand how to use arithmetic and combinatorial analysis to solve this interesting puzzle. Suitable for students in Grades 5-7.

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