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?

We need to calculate the probability that all top three places are taken by girls, and the last place is taken by a boy. Here's how we approach the problem:

### Step 1: Total number of ways to arrange all the participants
There are 6 girls and 4 boys, making a total of 10 participants. The total number of ways to arrange 10 participants in 10 places 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: Favorable arrangements
We want the top three places to be occupied by girls and the last place to be taken by a boy. The number of favorable outcomes can be broken into two parts:
- **Top 3 places:** The top 3 places should be filled by 3 girls out of the 6 girls. The number of ways to choose and arrange 3 girls from 6 is:
  \[
  P(6, 3) = \frac{6!}{(6-3)!} = 6 \times 5 \times 4 = 120
  \]

- **Last place:** The last place should be taken by one boy. There are 4 boys, so the number of ways to choose 1 boy for the last place is:
  \[
  4
  \]

- **Remaining 6 places:** The remaining 6 places (positions 4 to 9) are filled by the remaining 3 girls and 3 boys. The number of ways to arrange these 6 participants is:
  \[
  6! = 720
  \]

Thus, the total number of favorable outcomes is:
\[
120 \times 4 \times 720 = 345,600
\]

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

So, the probability is approximately \( \boxed{0.0952} \) or 9.52%.

Would you like more details on any step or any related questions? Here are 5 more questions you might find interesting:

1. What if we wanted only boys in the top three places?
2. How would the problem change if there were an equal number of boys and girls?
3. What if the top three places were all girls, but the last place could be anyone?
4. How would the probability change if we were to choose from a larger group of boys and girls?
5. Can we calculate the probability for different finishing positions (e.g., top 5 places)?

**Tip:** When dealing with probability problems involving arrangements, breaking the problem into smaller parts can simplify the solution.

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 of girls finishing in the top three places and a boy in the last place in a race with 6 girls and 4 boys. This problem covers key probability and combinatorics concepts, using permutation and probability formulas. Suitable for Grades 10-12.

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