Math Problem Statement
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.
Solution
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 team members (11 soccer, 7 swimming, and 5 judo members).
-
Choose 3 members out of 23:
The total number of ways to choose 3 members from 23 is given by the combination formula: -
Number of running orders for 3 members:
Once the members are chosen, they can be arranged in ways (since order matters): Therefore, the total number of ways to select 3 members and assign them a running order is:
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:
-
Choose 1 swimmer from 7:
The number of ways to choose 1 swimmer from 7 is: -
Choose 1 judo member from 5:
The number of ways to choose 1 judo member from 5 is: -
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: -
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:
Probability:
The probability is the ratio of favorable outcomes to the total possible outcomes: Rounding to the first decimal place, the probability is:
Would you like further details or clarification?
Here are 5 questions to deepen your understanding:
- How do combinations and permutations differ in this problem?
- What is the effect of increasing the number of team members on the probability?
- How would the probability change if we had 8 swimmers instead of 7?
- What is the role of the factorial function in calculating permutations?
- 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).
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Combinatorics
Permutations
Combinations
Probability
Formulas
Combination formula: C(n, k) = n! / (k!(n-k)!)
Permutation formula: P(n) = n!
Probability formula: P(event) = favorable outcomes / total outcomes
Theorems
Basic Counting Principle
Factorial and Permutation Theorem
Suitable Grade Level
Grades 10-12
Related Recommendation
Probability of Relay Order with Random Selection from Soccer, Swimming, and Judo Teams
Calculate Probability of Relay Order with Swimmers and Judo Athletes
Probability of Soccer Players Passing the Baton in a Random Relay
Probability of Relay Race Without Consecutive Same-Group Members
Calculating Relay Race Probability with Random Team Selection from Soccer and Swimming Teams