How Many Combinations Can You Make with 1, 2, 3, 4, and 5?
Let's explore the different ways we can arrange the numbers 1, 2, 3, 4, and 5.
Understanding Combinations vs. Permutations
First, we need to understand the difference between combinations and permutations.
- Combinations are about selecting items from a set without regard to order. For example, the combinations "1, 2, 3" and "3, 2, 1" are considered the same because they contain the same numbers.
- Permutations are about arranging items in a specific order. The permutations "1, 2, 3" and "3, 2, 1" are considered different because they have different arrangements.
Calculating Combinations
To calculate the number of combinations, we use the following formula:
nCr = n! / (r! * (n-r)!)
Where:
- n is the total number of items (in our case, 5)
- r is the number of items we are selecting (we can choose any number from 1 to 5)
- ! represents the factorial (e.g., 5! = 5 * 4 * 3 * 2 * 1)
Let's break down the combinations for different values of 'r':
- r = 1 (selecting one number):
- 5C1 = 5! / (1! * 4!) = 5
- There are 5 ways to choose one number from the set (1, 2, 3, 4, 5).
- r = 2 (selecting two numbers):
- 5C2 = 5! / (2! * 3!) = 10
- There are 10 ways to choose two numbers from the set.
- r = 3 (selecting three numbers):
- 5C3 = 5! / (3! * 2!) = 10
- There are 10 ways to choose three numbers from the set.
- r = 4 (selecting four numbers):
- 5C4 = 5! / (4! * 1!) = 5
- There are 5 ways to choose four numbers from the set.
- r = 5 (selecting all five numbers):
- 5C5 = 5! / (5! * 0!) = 1
- There is only 1 way to choose all five numbers from the set.
Calculating Permutations
To calculate the number of permutations, we use the following formula:
nPr = n! / (n-r)!
Let's calculate the permutations for different values of 'r':
- r = 1 (arranging one number):
- 5P1 = 5! / 4! = 5
- There are 5 ways to arrange one number from the set.
- r = 2 (arranging two numbers):
- 5P2 = 5! / 3! = 20
- There are 20 ways to arrange two numbers from the set.
- r = 3 (arranging three numbers):
- 5P3 = 5! / 2! = 60
- There are 60 ways to arrange three numbers from the set.
- r = 4 (arranging four numbers):
- 5P4 = 5! / 1! = 120
- There are 120 ways to arrange four numbers from the set.
- r = 5 (arranging all five numbers):
- 5P5 = 5! / 0! = 120
- There are 120 ways to arrange all five numbers from the set.
Conclusion
By understanding the difference between combinations and permutations, we can calculate the number of ways to arrange or select items from a set. In our case, we found that there are multiple ways to combine or arrange the numbers 1, 2, 3, 4, and 5, depending on how many items we choose or the order we arrange them in.