https://r2.erweima.ai/mathbot/image/1726047615026o7zw4phb.jpg

Let's break down the problem based on what is visible from the image.

### Problem 1: General Team Formation
We have 7 engineers and 13 technicians in a company, and we want to form intervention teams of 5 people. The question is:

**1. How many different teams can be formed?**

Here, you need to choose 5 people from a total of \( 7 + 13 = 20 \) people (7 engineers and 13 technicians). This is a simple combination problem, where the order of selection doesn't matter.

The formula for combinations is:

\[
C(n, r) = \frac{n!}{r!(n-r)!}
\]

Where \( n \) is the total number of people to choose from (20) and \( r \) is the number of people in each team (5).

Thus, the number of different teams is:

\[
C(20, 5) = \frac{20!}{5!(20-5)!} = \frac{20!}{5!15!}
\]

### Problem 2: Team Formation with Constraints

**2a. There must be exactly one engineer.**

In this case, you are choosing 1 engineer from the 7 available and 4 technicians from the 13 available. This is a combination of two separate choices:

\[
C(7, 1) \times C(13, 4)
\]

Where \( C(7, 1) \) is the number of ways to choose 1 engineer from 7, and \( C(13, 4) \) is the number of ways to choose 4 technicians from 13.

**2b. There must be at least one engineer.**

To find this, we first calculate how many teams can be formed with no engineers, and then subtract that from the total number of teams.

- Teams with no engineers: This is choosing all 5 people from the 13 technicians.

\[
C(13, 5)
\]

- Teams with at least one engineer: Subtract the number of teams with no engineers from the total number of teams.

\[
\text{Teams with at least one engineer} = C(20, 5) - C(13, 5)
\]

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Would you like me to compute these values explicitly?

### Further Questions to Explore:
1. How would the problem change if the teams were of 6 people instead of 5?
2. How many teams can be formed if there must be exactly two engineers?
3. What if we had only 10 technicians instead of 13?
4. How many teams can be formed if there must be exactly 3 engineers and 2 technicians?
5. Can we form a team where no technician is present at all?

#### Tip: 
Whenever constraints are imposed on selection problems, it's often helpful to break the task into smaller steps and then multiply the results together.

We have 7 engineers and 13 technicians in a company, we want to form intervention teams of 5 people. How many different teams can be formed? Same question if: a. There must be exactly one engineer. b. There must be at least one engineer.

Learn how to solve a combinatorics problem involving forming teams of 5 people from a group of 7 engineers and 13 technicians. Explore different scenarios with constraints like having exactly one engineer or at least one engineer in the team.

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