Podcast
Questions and Answers
In how many ways can a person wear a shirt or a t-shirt if he has 3 shirts and 4 t-shirts?
In how many ways can a person wear a shirt or a t-shirt if he has 3 shirts and 4 t-shirts?
7
In how many ways can a teacher choose one boy and one girl from a class of 15 boys and 10 girls?
In how many ways can a teacher choose one boy and one girl from a class of 15 boys and 10 girls?
150
In how many ways can a person go from city A to city C via city B if there are 4 roads from A to B and 5 roads from B to C?
In how many ways can a person go from city A to city C via city B if there are 4 roads from A to B and 5 roads from B to C?
20
In how many ways can a person move out of city B if there are 4 roads from A to B and 5 roads from B to C?
In how many ways can a person move out of city B if there are 4 roads from A to B and 5 roads from B to C?
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In how many ways can a teacher choose one student for the monitor post from a class of 15 boys and 10 girls?
In how many ways can a teacher choose one student for the monitor post from a class of 15 boys and 10 girls?
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In how many ways can a teacher choose two students for captain and vice-captain from a class of 15 boys and 10 girls?
In how many ways can a teacher choose two students for captain and vice-captain from a class of 15 boys and 10 girls?
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In how many ways can a person go from city A to city C if there are 4 roads from A to B, 5 roads from B to C, and 2 roads from A to C?
In how many ways can a person go from city A to city C if there are 4 roads from A to B, 5 roads from B to C, and 2 roads from A to C?
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In how many ways can 1st, 2nd, and 3rd prizes be distributed among 7 students?
In how many ways can 1st, 2nd, and 3rd prizes be distributed among 7 students?
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In how many ways can a flag consisting of 5 vertical strips be designed using one or all colors from red, yellow, and blue?
In how many ways can a flag consisting of 5 vertical strips be designed using one or all colors from red, yellow, and blue?
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Study Notes
Permutations and Combinations Overview
- Permutation refers to the arrangement of items in a specific sequence or order.
- Combination refers to the selection of items without regard to the order.
Basic Concepts
- A person can wear either a shirt or a t-shirt. If he has 3 shirts and 4 t-shirts, he has 7 total options.
Routes and Pathways
- From city A to city B, there are 4 roads; from city B to city C, there are 5 roads.
- The total number of ways to travel from city A to city C via city B is the product of the roads from A to B and B to C, which is (4 \times 5 = 20).
Choosing Students
- In a class with 15 boys and 10 girls, a teacher can choose one boy and one girl in (15 \times 10 = 150) ways.
- The total ways to select one student for a monitor position from 25 students (15 boys and 10 girls) is 25.
- For a captain and vice-captain role, the teacher can select 2 distinct students in (C(25, 2) = \frac{25!}{2!(25-2)!} = 300) ways.
Count of Paths and Selections
- Total roads for the journey from A to C through B include:
- 4 roads from A to B,
- 5 roads from B to C,
- 2 additional roads from A to C directly.
- To travel from B to C, the number of ways equals 5 (roads available).
Multiple Prize Distributions
- For distributing 1st, 2nd, and 3rd prizes among 7 students, the combinations depend on whether order matters. If it does (as in a race), that is a permutation scenario.
Chairman and Vice-Chairman Selection
- In a committee of 8 individuals, the selection of a chairman and vice-chairman (where one person cannot hold both positions) can be calculated as (P(8, 2) = 8 \times 7 = 56).
Flag Design Combinations
- To design a flag with 5 vertical strips using the colors red, yellow, or blue:
- Each strip can be any of the 3 colors; thus, the total combinations are (3^5 = 243) different flag designs.
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Description
This quiz covers the fundamental concepts of permutations and combinations, emphasizing the difference between arranging items and selecting them. It includes practical examples such as clothing choices, travel routes, and student selections to illustrate these concepts in real-life scenarios.
