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

To solve this problem, we need to calculate the probability that the top three finishers in the race are all girls. Here's how we can do that step by step:

### Step 1: Total possible outcomes
There are 6 girls and 4 boys, so 10 children in total. The top three places can be filled by any combination of 3 children from these 10. To find the total number of ways to choose 3 children out of 10, we use the combination formula:

\[
\text{Total possible outcomes} = \binom{10}{3} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120
\]

### Step 2: Favorable outcomes (only girls in the top three)
To have only girls in the top three places, we need to select 3 girls from the 6 available. The number of ways to choose 3 girls from 6 is:

\[
\text{Favorable outcomes} = \binom{6}{3} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20
\]

### Step 3: Probability
The probability is the ratio of favorable outcomes to total outcomes:

\[
\text{Probability} = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{20}{120} = \frac{1}{6}
\]

Thus, the probability that only girls will finish in the top three places is **\(\frac{1}{6}\)**.

Would you like more details on any step, or further clarification?

Here are 5 related questions to deepen your understanding:
1. What is the probability that exactly two girls will finish in the top three places?
2. How many different ways can the top three finishers be arranged if we care about the order?
3. If there are 7 girls and 5 boys, what is the probability that only girls will finish in the top three?
4. What is the probability that the top three places will include at least one boy?
5. How does the probability change if there are more boys than girls in the race?

**Tip:** When dealing with probability, always ensure you clearly distinguish between combinations (order doesn't matter) and permutations (order matters).

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

Calculate the probability that only girls finish in the top three places in a race with 6 girls and 4 boys. This problem involves combinatorics and probability, with a step-by-step solution suitable for students in Grades 9-10.

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