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 calculate the probability that only girls will finish in the top three places and a boy will finish in the last place, we need to consider two parts: the selection of the top three positions and the last position.

### Step 1: Probability that only girls finish in the top three places
There are 6 girls in total, and we want the top 3 positions to be taken by girls. The number of ways to choose and arrange 3 girls out of 6 is given by permutations because the order matters (who comes first, second, and third).

The number of permutations of 3 girls from 6 is:
\[
P(6, 3) = \frac{6!}{(6-3)!} = 6 \times 5 \times 4 = 120
\]

### Step 2: Probability that a boy finishes in the last place
There are 4 boys, and we want exactly one of them to finish in the last place. The number of ways to choose and place a boy in the last position is:
\[
4 \text{ (choices for the last position)}
\]

### Step 3: Total number of possible outcomes
The total number of ways to arrange all 10 children (6 girls and 4 boys) in a race is simply the number of permutations of 10 people:
\[
10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 3,628,800
\]

### Step 4: Probability calculation
Now, the favorable outcomes are the number of ways to arrange 3 girls in the top 3 positions and 1 boy in the last place, while the remaining 6 people (3 girls and 3 boys) can occupy the other positions. The number of favorable outcomes is:
\[
P(6, 3) \times 4 = 120 \times 4 = 480
\]

Thus, the probability is:
\[
\text{Probability} = \frac{480}{10!} = \frac{480}{3,628,800} = \frac{1}{7,560}
\]

### Final Answer:
The probability that only girls will finish in the top three places and a boy in the last place is:
\[
\frac{1}{7,560}
\]

Would you like further details or clarification on any part of the explanation?

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Here are some related questions:
1. How do you calculate the probability of selecting boys in the top places?
2. What is the probability of boys and girls alternating positions in the race?
3. How do you calculate the probability of a specific girl winning the race?
4. What is the probability that all boys finish after all girls?
5. How would the probability change if there were 5 boys and 5 girls?

**Tip:** When calculating probabilities, always make sure to account for both the favorable outcomes and the total number of possible outcomes.

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?

Calculate the probability that only girls will finish in the top three positions and a boy will finish in the last place in a race with 6 girls and 4 boys. This solution involves permutations and the fundamental principle of counting, suitable for students in Grades 10-12.

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