assume that you have a total of 10 people on the board: 2 out of state seniors, 3 in state seniors, 2 out of state non seniors, and 3 in state non seniors. university rules require taht at least one in state student and at least one senior hold one of the three offices. note that if individuals change offices thena  differnet selection exists. in how many ways can the officers be chosen while still conforming to university rules

We are tasked with determining the number of ways to select three officers from a group of 10 people, ensuring that at least one in-state student and at least one senior hold one of the offices.

### Step 1: Breakdown of the group
The group of 10 people consists of:

- 2 out-of-state seniors
- 3 in-state seniors
- 2 out-of-state non-seniors
- 3 in-state non-seniors

### Step 2: Total number of unrestricted ways to select 3 officers
If there were no restrictions, the total number of ways to choose 3 officers from 10 people would be:

\[
\binom{10}{3} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120
\]

### Step 3: Subtracting invalid cases
We are required to have at least one in-state student and at least one senior as officers. We will first calculate the number of selections that violate these rules and subtract them from the total.

#### Invalid Case 1: No in-state students
If there are no in-state students, we are selecting all 3 officers from the out-of-state students (2 out-of-state seniors + 2 out-of-state non-seniors = 4 people total). The number of ways to choose 3 officers from the 4 out-of-state students is:

\[
\binom{4}{3} = 4
\]

#### Invalid Case 2: No seniors
If there are no seniors, we are selecting all 3 officers from the non-seniors (2 out-of-state non-seniors + 3 in-state non-seniors = 5 people total). The number of ways to choose 3 officers from the 5 non-seniors is:

\[
\binom{5}{3} = \frac{5 \times 4 \times 3}{3 \times 2 \times 1} = 10
\]

#### Invalid Case 3: No in-state students and no seniors
In this case, we are selecting all 3 officers from the out-of-state non-seniors (2 people total). Since there are only 2 out-of-state non-seniors, we cannot select 3 officers from this group, so the number of ways to do this is 0.

### Step 4: Applying the inclusion-exclusion principle
Using inclusion-exclusion, the total number of invalid cases is:

\[
\text{Invalid cases} = (\text{No in-state students}) + (\text{No seniors}) - (\text{No in-state students and no seniors})
\]

\[
\text{Invalid cases} = 4 + 10 - 0 = 14
\]

### Step 5: Valid selections
The number of valid selections is the total number of ways to choose 3 officers minus the invalid cases:

\[
\text{Valid selections} = 120 - 14 = 106
\]

Thus, there are **106** valid ways to choose the officers while conforming to the university's rules.

---

Do you want more details on any part of this solution or have any questions?

Here are 5 relative questions to expand on this problem:

1. What if there were additional restrictions, such as a maximum number of non-seniors allowed as officers?
2. How would the result change if there were more people in each group?
3. What if the university rules also required at least one out-of-state student?
4. How does the inclusion-exclusion principle apply to different types of combinatorics problems?
5. What is the significance of the binomial coefficient in counting problems?

**Tip:** In combinatorics, always start by calculating the total number of possibilities, then subtract the invalid cases to find the valid ones.

Assume that you have a total of 10 people on the board: 2 out of state seniors, 3 in state seniors, 2 out of state non seniors, and 3 in state non seniors. University rules require that at least one in-state student and at least one senior hold one of the three offices. Note that if individuals change offices, then a different selection exists. In how many ways can the officers be chosen while still conforming to university rules?

Learn how to solve an officer selection problem involving combinatorics with constraints on in-state students and seniors. This problem covers key mathematical concepts like the inclusion-exclusion principle and binomial coefficients, suitable for high school and undergraduate students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation