There are 6 girls and 4
boys running a race.  What is the probability that only girls will finish in the top three places and a boy in the last place?

To find the probability that only girls will finish in the top three places and a boy will finish in the last place, we can break the problem into two parts: selecting the top three girls and placing a boy in the last position.

### Step-by-Step Solution:

1. **Total number of participants**: There are \(6\) girls and \(4\) boys, so the total number of participants is \(6 + 4 = 10\).

2. **Total number of ways to arrange all 10 participants**:
   The total number of ways to arrange 10 participants in the race is \(10!\) (factorial of 10), which represents every possible order the 10 runners can finish.

3. **Number of favorable outcomes**:
   - The top three positions must be occupied by girls. To choose and arrange 3 girls from the 6 available, the number of ways to do this is:
     \[
     \text{Ways to select and arrange 3 girls} = \binom{6}{3} \times 3! = \frac{6!}{(6-3)!3!} \times 3! = 20 \times 6 = 120
     \]
   - A boy must finish in the last place. Since there are 4 boys to choose from, the number of ways to place a boy in the last position is \(4\).
   - The remaining 6 positions (places 4 to 9) can be filled by the remaining 3 girls and 3 boys. There are \(6!\) ways to arrange these 6 people.

4. **Total number of favorable outcomes**:
   The total number of favorable outcomes is the product of the ways to:
   - Arrange 3 girls in the top three places: \(120\),
   - Choose a boy for the last place: \(4\),
   - Arrange the remaining 6 participants in the middle positions: \(6!\).

   Hence, the number of favorable outcomes is:
   \[
   \text{Favorable outcomes} = 120 \times 4 \times 6! = 120 \times 4 \times 720 = 345,600
   \]

5. **Total number of possible outcomes**:
   The total number of possible outcomes, which is the number of ways to arrange all 10 participants, is \(10!\):
   \[
   \text{Total outcomes} = 10! = 3,628,800
   \]

6. **Probability**:
   The probability is the ratio of favorable outcomes to total outcomes:
   \[
   \text{Probability} = \frac{345,600}{3,628,800} = \frac{1}{10.5} \approx 0.0952
   \]

So, the probability is approximately **0.0952** or **9.52%**.

---

Would you like any further details on this problem?

Here are 5 related questions to consider:

1. What is the probability that only boys will finish in the top three places?
2. How does the probability change if there are 5 girls and 5 boys?
3. What if the condition was that only girls finish in the top two places instead of the top three?
4. What is the probability that a boy finishes first and a girl finishes last?
5. How would the answer change if no restrictions were placed on the middle positions?

**Tip**: Understanding permutations and combinations can help solve more complex probability problems involving specific positions or sequences!

There are 6 girls and 4 boys running a race. What is the probability that only girls will finish in the top three places and a boy in the last place?

Learn how to calculate the probability that only girls finish in the top three places and a boy finishes last in a race with 6 girls and 4 boys. This problem involves key concepts of probability, permutations, and combinations, providing a step-by-step guide suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation