Are you curious about how many 3-digit combinations exist?

If you’re short on time, here’s a quick answer to your question: **There are 900 3-digit combinations.**

In this article, we will dive deeper into the topic and explore the reasoning behind this number.

## Understanding the Basics

When it comes to understanding combinations, it’s important to first define what a combination is. A combination is a way of selecting items from a larger set without regard to the order in which the items are chosen. This is different from a permutation, which takes into account the order in which the items are chosen.

A 3-digit combination, as the name suggests, consists of three digits chosen from the set of digits 0 through 9. This means that there are a total of 1,000 possible 3-digit combinations (10 options for the first digit, 10 options for the second digit, and 10 options for the third digit).

But why is order important in permutations but not in combinations? Consider the example of choosing a president, vice president, and secretary from a group of 10 people. If we are selecting a permutation, we care about the order in which the positions are filled. This means that there are 10 options for the president, 9 options for the vice president (since we’ve already chosen one person), and 8 options for the secretary (since we’ve chosen two people). This gives us a total of 720 possible permutations.

However, if we are selecting a combination, we don’t care about the order in which the positions are filled. This means that there are still 10 options for the president, but only 9 options for the vice president (since we’ve already chosen one person) and 8 options for the secretary (since we’ve chosen two people). This gives us a total of 120 possible combinations.

- A 3-digit combination consists of three digits chosen from the set of digits 0 through 9, resulting in 1,000 possible combinations.
- Order is important in permutations but not in combinations, as permutations take into account the order in which items are chosen while combinations do not.

## Calculating the Number of 3-Digit Combinations

When it comes to calculating the number of 3-digit combinations, there are four steps to follow:

**Step 1:**Determine the number of choices for the first digit**Step 2:**Determine the number of choices for the second digit**Step 3:**Determine the number of choices for the third digit**Step 4:**Multiply the results from Steps 1-3 to obtain the total number of 3-digit combinations

Let’s take a closer look at each step:

### Step 1: Determine the number of choices for the first digit

The first digit in a 3-digit combination can be any number from 1 to 9. This gives us a total of 9 choices for the first digit.

### Step 2: Determine the number of choices for the second digit

Once we have chosen a first digit, we need to determine the number of choices for the second digit. We can choose any number from 0 to 9 for the second digit, including the number we chose for the first digit. This gives us a total of 10 choices for the second digit.

### Step 3: Determine the number of choices for the third digit

Finally, we need to determine the number of choices for the third digit. Like the second digit, we can choose any number from 0 to 9 for the third digit, including the numbers we chose for the first and second digits. This also gives us a total of 10 choices for the third digit.

### Step 4: Multiply the results from Steps 1-3 to obtain the total number of 3-digit combinations

To obtain the total number of 3-digit combinations, we need to multiply the results from Steps 1-3. This gives us:

Total number of choices for the first digit | 9 |

Total number of choices for the second digit | 10 |

Total number of choices for the third digit | 10 |

Total number of 3-digit combinations | 9 x 10 x 10 = 900 |

Therefore, there are 900 possible 3-digit combinations.

Calculating the number of 3-digit combinations is a simple but important mathematical problem that has many applications in fields such as computer science, cryptography, and statistics. For more information on permutations and combinations, you can visit Math is Fun.

## Exploring the Concept Further

So, we know that there are 900 possible 3-digit combinations if repetition is not allowed and order matters. But what if repetition is allowed? In this case, we can have any number from 000 to 999, which gives us a total of **1000** possible combinations.

Now, let’s say that we cannot repeat certain digits. For example, we cannot have a combination with two 5s. In this case, we need to calculate the number of ways we can select the digits for each place value. For the first digit, we have 9 options (we cannot choose 0 or 5). For the second digit, we have 8 options (we cannot choose the digit we chose for the first digit or 5). And for the third digit, we have 7 options (we cannot choose the digits we chose for the first two digits or 5). Therefore, the total number of possible combinations is 9 x 8 x 7 = **504**.

Finally, what if the order doesn’t matter? In other words, we’re looking for the number of distinct sets of 3 digits. To calculate this, we can use the formula for combinations, which is nCr = n! / r!(n-r)!, where n is the total number of items and r is the number of items we want to choose. In this case, n = 10 (since we have 0-9 to choose from) and r = 3. So, the number of distinct sets of 3 digits is 10C3 = 10! / 3!(10-3)! = **120**.

It’s important to note that the number of possible combinations can vary depending on the specific rules and restrictions applied. Understanding these nuances can help us better understand and appreciate the beauty of mathematics.

## Conclusion

In conclusion, there are 900 possible 3-digit combinations when repetition is not allowed and order matters.

Combinations can be a fascinating topic to explore and can have practical applications in fields such as mathematics, computer science, and cryptography.

We hope this article has helped you better understand the concept of combinations and provided an answer to your question.