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 this down step-by-step.

### Step 1: Total number of possible outcomes
There are a total of 10 participants (6 girls and 4 boys). The total number of ways to arrange these 10 participants in a race 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 outcomes
We want only girls in the top three places and a boy in the last place.

#### Top 3 places:
- The number of ways to select 3 girls from 6 is \( \binom{6}{3} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20 \).
- The number of ways to arrange those 3 girls in the top 3 places is \( 3! = 6 \).

So, the number of ways to arrange the top 3 places with girls is \( 20 \times 6 = 120 \).

#### Last place:
- The number of ways to select 1 boy for the last place from 4 boys is \( 4 \).

#### Remaining 6 places:
- After choosing the top 3 places and the last place, we have 6 participants left (3 girls and 3 boys). These 6 can be arranged in any order in the remaining 6 spots, and the number of ways to arrange them is \( 6! = 720 \).

### Step 3: Total favorable outcomes
The total number of favorable outcomes is the product of all the individual steps:

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

### Step 4: Probability
The probability is the ratio of favorable outcomes to the total outcomes:

\[
P = \frac{345,600}{3,628,800} = \frac{1}{10.5} = \frac{2}{21}
\]

### Final Answer:
The probability that only girls will finish in the top three places and a boy will finish in the last place is:

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

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Would you like further clarification or details on any part of this solution?

Here are some related questions to explore:
1. What is the probability that all boys finish before any girl in the race?
2. What is the probability that a specific boy finishes in first place?
3. How many ways can the boys and girls be arranged in alternate positions?
4. What is the probability that a girl finishes first and a boy finishes last?
5. How many distinct ways can all the participants finish the race?

**Tip:** When calculating probabilities, breaking down the problem into smaller, manageable parts can simplify the process.

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?

Explore the probability that 6 girls and 4 boys running a race will result in only girls finishing in the top three positions and a boy in the last place. The solution involves combinatorics, factorials, and probability theory, making it suitable for high school students in Grades 10-12.

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