There are 10,000 combinations of 4 single-digit numbers when repetition is allowed.
The question can be interpreted in a few ways, so let's explore different scenarios and their answers:
Scenario 1: Single-Digit Numbers with Repetition Allowed
When we talk about combinations using single-digit numbers, we typically mean using the digits 0 through 9. If we are allowed to repeat numbers, each of the four positions can be filled by any of these 10 digits.
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Understanding the Counting Principle: The Counting Principle states that if you have 'm' ways to do one thing and 'n' ways to do another, then you have m * n ways to do both. This extends to any number of events.
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Calculation: For a 4-digit sequence where each position can be any of the 10 digits (0-9), the total combinations are:
10 10 10 * 10 = 104 = 10,000 -
Example: Possible combinations include 0000, 1234, 9999, 2020, etc.
Scenario 2: Single-Digit Numbers Without Repetition Allowed
If no repetition is allowed, the calculation changes.
- Understanding No Repetition: Once a number is used in one position, it can't be used again in another position.
- Calculation: For the first position we have 10 choices, for the second 9 (since one number has already been used), for the third 8, and for the fourth 7. Therefore we have 10987 combinations.
10 9 8 7 = 5,040 - Example: 1234 is valid, but 1232 is not because 2 is repeated.
Scenario 3: Combinations of Single Numbers (Order Doesn't Matter, Without Repetition)
This scenario refers to combinations in the mathematical sense, where the order of the numbers does not matter. However, the question specifies "4 single numbers," so it is likely a reference to permutations.
- Understanding Combinations: Combinations are selections where order is irrelevant. For example, {1, 2, 3, 4} is the same combination as {4, 3, 2, 1}.
- Calculation: To calculate combinations (without repetition and where order doesn't matter) of 4 single digits from 10 possible digits we would use the formula nCr = n! / (r! (n-r)!).
In our case, this is 10C4 = 10! / (4! (10-4)!), which calculates to 210.
10! / (4! 6!) = (10987) / (432*1) = 210
Summary Table
| Scenario | Repetition Allowed? | Order Matters? | Combinations |
|---|---|---|---|
| 4 single digits (0-9) | Yes | Yes | 10,000 |
| 4 single digits (0-9) | No | Yes | 5,040 |
| Combinations of 4 single digits (0-9) | No | No | 210 |
Based on the most common interpretation of the question, which implies repetition is allowed, there are 10,000 combinations of 4 single numbers.
