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 find the probability that only girls will finish in the top three places and a boy will finish in the last place.

### Step 1: Total number of ways to arrange all 6 girls and 4 boys

Since there are 6 girls and 4 boys, the total number of ways to arrange all 10 children in the race is simply the number of permutations of 10 distinct individuals:
\[
10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 3,628,800
\]

### Step 2: Number of favorable outcomes

To calculate the number of favorable outcomes, we need to arrange:

1. The top 3 places should be filled by **girls only**.
2. The last place (10th) must be filled by a **boy**.
3. The remaining 6 positions (from 4th to 9th) can be filled by the remaining 3 boys and 3 girls.

#### Step 2.1: Arranging the top 3 places with girls
We have 6 girls, and we need to choose 3 of them for the top 3 positions. This can be done in:
\[
\text{Ways to arrange 3 girls from 6} = P(6, 3) = \frac{6!}{(6-3)!} = 6 \times 5 \times 4 = 120
\]

#### Step 2.2: Arranging a boy in the last place
Since there are 4 boys, and one boy needs to be placed in the last position (10th place), there are 4 possible choices for the last place.

#### Step 2.3: Arranging the remaining 6 children in the middle 6 positions
Now, we have 3 boys and 3 girls left, and we need to arrange them in the 6 remaining positions (4th to 9th places). This can be done in:
\[
\text{Ways to arrange 6 remaining children} = 6! = 720
\]

### Step 3: Total number of favorable outcomes
The total number of favorable outcomes is the product of the three steps:
\[
120 \times 4 \times 720 = 345,600
\]

### Step 4: Probability
The probability is the ratio of favorable outcomes to the total number of outcomes:
\[
\text{Probability} = \frac{345,600}{3,628,800} = \frac{24}{252} = \frac{2}{21}
\]

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 a more detailed breakdown or further clarification?

Here are 5 related questions for further exploration:
1. What is the probability that only girls finish in the top two places?
2. How many ways can the boys and girls be arranged if no restrictions are placed?
3. What is the probability that a girl finishes in the first place and a boy finishes in the last place?
4. How would the probability change if there were 7 girls and 3 boys?
5. What if we required that only boys finish in the top three places?

**Tip**: When calculating probabilities, breaking the problem into smaller parts and calculating them step-by-step can make it easier to keep track of all possibilities.

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 problem where 6 girls and 4 boys are running a race. Learn how to calculate the probability that only girls finish in the top three places and a boy finishes in the last place using permutations and the probability formula. Suitable for high school students.

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