When dealing with combinations, the number of possible combinations depends on whether repetition is allowed and whether the order of the numbers matters. Let’s explore the different scenarios:
1. Combinations without Repetition and Order Matters (Permutations)
If you have 3 distinct numbers and the order matters, the number of possible combinations (permutations) is calculated using the formula for permutations:
P(n, k) = n! / (n – k)!
Where:
- n is the total number of items.
- k is the number of items to choose.
For 3 numbers, n = 3 and k = 3:
P(3, 3) = 3! / (3 – 3)! = 6 / 1 = 6
So, there are 6 possible permutations.
2. Combinations without Repetition and Order Does Not Matter
If the order does not matter, the number of combinations is calculated using the combination formula:
C(n, k) = n! / (k! * (n – k)!)
For 3 numbers, n = 3 and k = 3:
C(3, 3) = 3! / (3! * (3 – 3)!) = 6 / (6 * 1) = 1
So, there is only 1 possible combination.
3. Combinations with Repetition and Order Matters
If repetition is allowed and the order matters, the number of combinations is calculated as:
n^k
For 3 numbers, n = 3 and k = 3:
3^3 = 27
So, there are 27 possible combinations.
4. Combinations with Repetition and Order Does Not Matter
If repetition is allowed and the order does not matter, the number of combinations is calculated using the formula:
C(n + k – 1, k)
For 3 numbers, n = 3 and k = 3:
C(3 + 3 – 1, 3) = C(5, 3) = 10
So, there are 10 possible combinations.
In summary, the number of possible combinations with 3 numbers can vary depending on the conditions:
- 6 if order matters and no repetition.
- 1 if order does not matter and no repetition.
- 27 if order matters and repetition is allowed.
- 10 if order does not matter and repetition is allowed.