suppose that 10 female and 12 male applicants have been successfully screened for 8 positions. In how many ways can the following compositions be selected? at least 7 males

To solve this problem, we need to determine how many ways we can select 8 applicants such that at least 7 of them are male, from a group of 10 female and 12 male applicants.

### Breakdown of "At Least 7 Males"
The phrase "at least 7 males" means that we need to consider the following two scenarios:
1. **Exactly 7 males and 1 female**.
2. **Exactly 8 males and 0 females**.

### Step 1: Calculate for 7 Males and 1 Female
- We select 7 males from the 12 male applicants. The number of ways to do this is given by the combination \( C(12, 7) \).
- We select 1 female from the 10 female applicants. The number of ways to do this is given by the combination \( C(10, 1) \).

The total number of ways to select exactly 7 males and 1 female is the product of these two combinations:
\[
\text{Ways for 7 males and 1 female} = C(12, 7) \times C(10, 1)
\]

### Step 2: Calculate for 8 Males and 0 Females
- We select 8 males from the 12 male applicants. The number of ways to do this is given by the combination \( C(12, 8) \).

Since no females are selected, the number of ways to select 8 males and 0 females is:
\[
\text{Ways for 8 males and 0 females} = C(12, 8)
\]

### Step 3: Sum the Two Scenarios
Finally, to get the total number of ways to select at least 7 males, we add the results of the two cases:
\[
\text{Total ways} = \left( C(12, 7) \times C(10, 1) \right) + C(12, 8)
\]

Let's compute this.# Calculate combinations for the two scenarios
ways_7_males_1_female = math.comb(12, 7) * math.comb(10, 1)
ways_8_males_0_females = math.comb(12, 8)

# Total number of ways to select at least 7 males
total_ways = ways_7_males_1_female + ways_8_males_0_females
total_waysThe total number of ways to select 8 applicants with at least 7 males from 10 female and 12 male applicants is 8,415.

Would you like further clarification on this calculation or have any other questions?

Here are some related questions:
1. How would the result change if we required at least 6 males instead of 7?
2. How do you calculate combinations when the number of positions exceeds the number of male applicants?
3. What if we wanted to select exactly 6 males and 2 females?
4. How does the inclusion of gender constraints impact selection probabilities?
5. How would the answer change if the number of applicants were different (e.g., 10 males and 12 females)?
6. What is the probability of selecting exactly 7 males out of 8 positions?
7. How can these techniques be applied to more complex selection problems?
8. What if we were interested in the opposite scenario, selecting at least 7 females?

**Tip:** When solving "at least" problems, remember to break down the problem into different cases, and add up the number of valid combinations for each case.

Learn how to calculate the number of ways to select at least 7 males from a group of 10 female and 12 male applicants for 8 positions. Detailed step-by-step solution using combinatorial mathematics. Suitable for advanced high school students and anyone interested in combinatorial problems.

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