Combinations And Permutations
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Combinations vs. Permutations
- Combinations: The number of ways to choose a sample of 'r' elements from a set of 'n' distinct objects where the order of selection does not matter.
- Permutations: The number of ways to arrange a sample of 'r' elements from a set of 'n' distinct objects where the order of selection is important.
Combinations & Permutations
The Mathematics of Ordering and Selecting.
What are Permutations and Combinations?
Permutations and combinations are fundamental concepts in combinatorics, a branch of mathematics concerned with counting, arrangement, and selection.
They provide a precise way to count the number of possible outcomes in various situations.
The key difference between them lies in one simple question: Does the order of selection matter?
Example:Imagine you have three letters: A, B, C. How many ways can you arrange them? How many ways can you choose a group of two letters? These questions are answered by permutations and combinations.
Permutations: When Order Matters
A permutation is an arrangement of objects in a specific order. Think of it as a 'line-up' or a 'sequence'.
When we talk about permutations, changing the order of the objects creates a new and distinct outcome.
Formula: The number of permutations of 'r' objects taken from a set of 'n' objects is given by P(n, r) = n! / (n-r)!
Example:A race with 8 runners. The number of ways to award gold, silver, and bronze medals is a permutation because the order (1st, 2nd, 3rd) is crucial. (ABC is different from BAC).
Combinations: When Order Doesn't Matter
A combination is a selection of objects where the order of selection is not important. Think of it as a 'group' or a 'committee'.
When we talk about combinations, ABC is the same group as BAC, CAB, etc. We only care about which objects are chosen, not the order in which they were chosen.
Formula: The number of combinations of 'r' objects taken from a set of 'n' objects is C(n, r) = n! / (r!(n-r)!).
Example:Choosing 3 people from a group of 8 to form a committee is a combination. A committee of Ann, Bob, and Chris is the same regardless of who was picked first.
The Factorial (!)
Both formulas use the factorial function, denoted by an exclamation mark (!).
The factorial of a non-negative integer 'n', denoted by n!, is the product of all positive integers less than or equal to n.
For example, 5! = 5 × 4 × 3 × 2 × 1 = 120.
By definition, 0! = 1.
Example:To calculate 4!, you would multiply 4 * 3 * 2 * 1 = 24.
Real-World Application: Probability and Security
These concepts are critical in many fields.
Probability: They are used to calculate the number of possible outcomes, which is the basis of finding probabilities (e.g., the odds of winning the lottery).
Computer Science: Used in algorithms for sorting, cryptography, and network routing.
Security: The number of possible passwords of a certain length is a permutation with repetition, which helps in understanding password strength.
Example:Lottery numbers are a combination because the order in which the numbers are drawn doesn't matter. You win as long as you have the correct group of numbers.
Key Summary
- **Permutation:** Order matters. Think arranging, ranking, or assigning specific roles.
- **Combination:** Order does not matter. Think choosing a group, committee, or sample.
- The **factorial (!)** is crucial for calculating both.
- Always ask: "Is the order important?" to decide which method to use.
Practice Problems
Problem: From a class of 20 students, how many different ways can a president, vice president, and treasurer be elected?
Ask yourself: does order matter? Yes, because the positions are distinct. This is a permutation. Here n=20 and r=3.
Solution: P(20, 3) = 20! / (20-3)! = 20! / 17! = 20 × 19 × 18 = 6,840 ways.
Problem: You have 10 books and want to choose 4 to take on vacation. How many different groups of 4 books can you choose?
Does order matter? No, you just care about the group of books, not the order you picked them in. This is a combination. Here n=10 and r=4.
Solution: C(10, 4) = 10! / (4!(10-4)!) = 10! / (4!6!) = (10×9×8×7) / (4×3×2×1) = 210 different groups.
Problem: How many unique 5-letter 'words' can be formed from the letters in 'APPLE'?
This is a permutation of 5 letters. However, since the letter 'P' is repeated, we must divide by the factorial of the number of repetitions to avoid overcounting.
Solution: Total arrangements = 5! / 2! = 120 / 2 = 60 unique words.
Frequently Asked Questions
How can I easily remember the difference?
Think 'P' for Permutation and 'Position'. If the position/order matters, it's a permutation. Think 'C' for Combination and 'Committee'. For a committee, the order you pick people in doesn't matter.
What if repetition is allowed?
The formulas change. If repetition is allowed, the number of permutations is n^r. For example, a 3-digit lock (0-9) has 10³ = 1000 possible permutations (000 to 999).
Why is 0! equal to 1?
It's a mathematical convention that makes many formulas, including the combination and permutation formulas, work correctly. One way to think of it is that there is exactly one way to arrange zero objects (by doing nothing).
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