How many combinations can be made with 3 digits?
Understand the Problem
The question is asking for the total number of possible combinations that can be formed using 3 digits. This can involve different scenarios, such as if digits can repeat or not, and if leading zeros are allowed.
Answer
- Without leading zeros and without repetition: 648 - With leading zeros and without repetition: 720 - With leading zeros and repetition allowed: 1000
Answer for screen readers
- Without leading zeros and without repetition: 648 combinations
- With leading zeros and without repetition: 720 combinations
- With leading zeros and repetition allowed: 1000 combinations
Steps to Solve
- Identify the scenarios for digit combinations
We need to clarify if digits can repeat and if leading zeros are allowed. For this solution, let's explore both scenarios:
- Without leading zeros and without repetition: The first digit can be any from 1-9, and the other two can be any from 0-9 excluding the chosen digits.
- With leading zeros and without repetition: Any of the three digits can be 0-9, with each digit being distinct.
- With leading zeros and repetition allowed: Each digit can be 0-9, and they can repeat.
- Calculate combinations without leading zeros and without repetition
For the first digit, we have 9 options (1-9).
For the second digit, we have 10 options (0-9) but can't repeat the first digit, so 9 options left.
For the third digit, we can't repeat the first two digits, so that gives us 8 options.
The total combinations would be calculated as: $$ 9 \times 9 \times 8 = 648 $$
- Calculate combinations with leading zeros and without repetition
In this scenario, each digit can be any from 0-9, but they can't repeat.
So for the first digit, there are 10 options (0-9).
For the second digit, there are 9 options (excluding the first digit),
And for the third digit, there are 8 options.
The calculation will be: $$ 10 \times 9 \times 8 = 720 $$
- Calculate combinations with repetition allowed and leading zeros
Here, each of the three digits can be any digit from 0-9, and they can repeat.
Thus, for each digit, we have 10 choices.
The total combinations are: $$ 10 \times 10 \times 10 = 1000 $$
- Without leading zeros and without repetition: 648 combinations
- With leading zeros and without repetition: 720 combinations
- With leading zeros and repetition allowed: 1000 combinations
More Information
The answer varies based on the conditions applied to the digit combinations. It's important to consider the contexts of repetition and leading zeros as they significantly impact the total number of combinations possible.
Tips
- Forgetting to account for leading zeros can lead to incorrect results, especially if the scenario allows for their use.
- Not distinguishing between scenarios where repetition is allowed versus not can also affect the counts significantly.
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