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 solve this problem, we need to calculate the probability that:

1. The top three positions are occupied by only girls.
2. The last position is occupied by a boy.

### Step 1: Total number of ways to arrange all runners
There are a total of 6 girls and 4 boys, making 10 runners in total. The number of possible ways to arrange all 10 runners is:

\[
10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 3,628,800
\]

### Step 2: Number of favorable outcomes
We want to find how many arrangements satisfy the condition that the top three positions are taken by girls and the last position is taken by a boy.

- **Top three positions**: Choose 3 girls from 6 to occupy the top three positions. The number of ways to do this is \( \binom{6}{3} \), which is the number of ways to select 3 girls from 6, and then we multiply it by \(3!\) to account for all the possible arrangements of these 3 girls:

\[
\binom{6}{3} \times 3! = \frac{6!}{(6-3)!3!} \times 3! = 20 \times 6 = 120
\]

- **Last position**: Choose 1 boy from 4 boys to occupy the last position. There are 4 ways to do this.

- **Remaining positions**: For the remaining 6 runners (3 girls and 3 boys left), there are \(6!\) possible ways to arrange them in the remaining 6 positions:

\[
6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720
\]

Thus, the number of favorable outcomes is:

\[
120 \times 4 \times 720 = 345,600
\]

### Step 3: Calculate the probability
The probability is the ratio of the number of favorable outcomes to the total number of possible outcomes:

\[
\text{Probability} = \frac{345,600}{3,628,800} = \frac{12}{126} = \frac{2}{21}
\]

Thus, the probability that only girls finish in the top three places and a boy finishes in the last place is:

\[
\boxed{\frac{2}{21}}
\]

Would you like further details or have any questions about this? Here are some related questions:

1. What is the probability that all the top three finishers are boys?
2. What is the probability that a boy finishes in the top position?
3. How many different ways can the boys and girls be arranged in the race?
4. What is the probability that the first and last positions are both occupied by girls?
5. What is the probability that exactly one girl finishes in the top three?

**Tip:** In probability, it's often useful to break down complex events into smaller, more manageable parts, as we did here by considering the top three and last place separately.

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 covers key probability concepts, combinatorics, and factorial calculations. Suitable for high school students in Grades 9-12.

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