Three members are chosen from 11 members of the soccer team, 7 members of the swimming team, and 5 members of the judo team to run in a relay race.What is the probability that the order in which the swimmers pass the baton directly to the judo team members when both the combination and the running order are chosen at random?Answer using whole numbers only, rounding to the first decimal place.

To solve this problem, we need to calculate the probability that swimmers pass the baton directly to judo team members, under the condition that both the combination of members and the running order are chosen at random.

### Total Number of Ways to Choose the 3 Members and Their Running Order:

We have a total of \( 11 + 7 + 5 = 23 \) team members (11 soccer, 7 swimming, and 5 judo members).

1. **Choose 3 members out of 23**:  
   The total number of ways to choose 3 members from 23 is given by the combination formula:
   \[
   \text{Total combinations} = \binom{23}{3} = \frac{23 \times 22 \times 21}{3 \times 2 \times 1} = 1771
   \]

2. **Number of running orders for 3 members**:  
   Once the members are chosen, they can be arranged in \( 3! \) ways (since order matters):
   \[
   3! = 6
   \]
   Therefore, the total number of ways to select 3 members and assign them a running order is:
   \[
   1771 \times 6 = 10626
   \]

### Favorable Outcome: Swimmer Passes the Baton Directly to a Judo Member

For this condition, we need exactly one swimmer and one judo member out of the 3 chosen members. The third member can be from any team (soccer, swimming, or judo). Here’s how to calculate the favorable outcomes:

1. **Choose 1 swimmer from 7**:  
   The number of ways to choose 1 swimmer from 7 is:
   \[
   \binom{7}{1} = 7
   \]

2. **Choose 1 judo member from 5**:  
   The number of ways to choose 1 judo member from 5 is:
   \[
   \binom{5}{1} = 5
   \]

3. **Choose 1 remaining member from the remaining 21 members**:  
   After selecting 1 swimmer and 1 judo member, 21 members are left. We can choose 1 remaining member from these 21:
   \[
   \binom{21}{1} = 21
   \]

4. **Number of ways to arrange the 3 members with the swimmer passing the baton to the judo member**:  
   The swimmer must pass the baton directly to the judo member, so the arrangement is fixed for these two positions. The remaining member can be placed in either the first or third position, giving us 2 possible arrangements.

Thus, the number of favorable outcomes is:
\[
7 \times 5 \times 21 \times 2 = 1470
\]

### Probability:

The probability is the ratio of favorable outcomes to the total possible outcomes:
\[
\text{Probability} = \frac{1470}{10626} \approx 0.1384
\]
Rounding to the first decimal place, the probability is:
\[
\boxed{0.1}
\]

Would you like further details or clarification?

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Here are 5 questions to deepen your understanding:
1. How do combinations and permutations differ in this problem?
2. What is the effect of increasing the number of team members on the probability?
3. How would the probability change if we had 8 swimmers instead of 7?
4. What is the role of the factorial function in calculating permutations?
5. How can we generalize this problem to more complex team scenarios?

**Tip**: When calculating probabilities, always ensure to distinguish between combinations (order doesn't matter) and permutations (order matters).

Three members are chosen from 11 members of the soccer team, 7 members of the swimming team, and 5 members of the judo team to run in a relay race. What is the probability that the order in which the swimmers pass the baton directly to the judo team members when both the combination and the running order are chosen at random? Answer using whole numbers only, rounding to the first decimal place.

Calculate the probability of a swimmer passing the baton directly to a judo member in a relay race. This problem involves combinatorics, including permutations and combinations, and is suitable for high school students. Learn how to use factorials and probability formulas to solve complex team selection problems.

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