How Many Different Combinations Of 3 Numbers

How Many Different Combinations of 3 Numbers Are There? Unlocking the World of Permutations and Combinations

This seemingly simple question opens the door to a fascinating area of mathematics: combinatorics. The answer, however, depends on a crucial detail: do the order of the numbers matter, and are repetitions allowed? This article will explore the different scenarios and provide you with the formulas and methods to calculate the possibilities.

Understanding the Key Differences: Permutations vs. Combinations

Before we dive into the calculations, let's clarify the difference between permutations and combinations. This distinction is critical in determining the correct approach.

• Permutations: The order of the numbers matters. For example, 123 is considered different from 321.
• Combinations: The order of the numbers does not matter. 123 is considered the same as 321.

Scenario 1: Permutations with Repetition Allowed

Let's assume we can choose from a set of 'n' numbers, and we can reuse numbers. For example, we might choose from the digits 0-9 (n=10) and create a 3-digit combination like 555. The formula for permutations with repetition is:

n^r

Where:

• n = the number of choices (e.g., 10 digits)
• r = the number of selections (e.g., 3 numbers)

In our example (3-digit combinations with repetition from 0-9), the calculation would be:

10³ = 1000

Therefore, there are 1000 different 3-digit combinations if repetition is allowed.

Scenario 2: Permutations without Repetition Allowed

Now, let's consider the scenario where we can't repeat numbers. This is often referred to as a permutation without replacement. The formula is:

n! / (n-r)!

Where:

• n! (n factorial) means n*(n-1)(n-2)...*1
• n = the number of choices
• r = the number of selections

If we choose 3 digits from 0-9 without repetition, the calculation is:

10! / (10-3)! = 10 * 9 * 8 = 720

There are 720 different 3-digit combinations if repetition is not allowed.

Scenario 3: Combinations with Repetition Allowed

Combinations are trickier when repetition is allowed. The formula involves combinations and is as follows:

(n+r-1)! / (r! * (n-1)!)

Where:

• n = the number of choices
• r = the number of selections

Using our 0-9 digit example:

(10+3-1)! / (3! * (10-1)!) = 220

This means there are 220 unique combinations of 3 digits if repetition is allowed and order doesn't matter.

Scenario 4: Combinations without Repetition Allowed

This is the simplest case. The formula is:

n! / (r! * (n-r)!)

Using our 10 digit example and selecting 3 without repetition:

10! / (3! * (10-3)!) = 120

This gives us 120 unique combinations.

In Summary:

The number of different combinations of 3 numbers drastically changes depending on whether repetition is allowed and if the order matters. Remember to carefully consider these factors before applying the appropriate formula. Understanding permutations and combinations is essential not only for solving mathematical puzzles but also for various applications in probability, statistics, and computer science.