When working with a set of three numbers, the number of combinations that can be made depends on whether repetition is allowed and the order matters. Let’s explore the different scenarios:

1. Combination without repetition, order doesn’t matter

In this case, each number can only appear once in a combination, and the order of the numbers doesn’t matter. To calculate the number of combinations, we can use the formula for combinations without repetition, given by:

C(n, r) = n! / (r!(n-r)!)

where n is the total number of elements (3 in our case) and r is the number of elements we want to select at a time (also 3).

Calculating it for our case: C(3, 3) = 3! / (3!(3-3)!) = 3! / (3! * 0!) = 1.

This means that there is only one combination without repetition and where the order doesn’t matter when working with a set of three numbers.

2. Combination without repetition, order matters

In this scenario, each number can only appear once, but the order of the numbers now matters. To calculate the number of combinations, we can simply use the formula for permutations without repetition, given by:

P(n, r) = n!

where n is the total number of elements and r is the number of elements we want to select at a time. In our case, P(3, 3) = 3! = 3 * 2 * 1 = 6.

With three numbers, there are six different combinations without repetition, but where the order matters.

3. Combination with repetition, order doesn’t matter

Sometimes, repetition is allowed within a combination. In this case, the order doesn’t matter. To calculate the number of combinations with repetition, we can use the formula:

C(n + r – 1, r) = (n + r – 1)! / ((n – 1)! * r!)

where n is the number of distinct elements (3 in our case) and r is the number of elements selected at a time (also 3).

Calculating, C(3 + 3 – 1, 3) = 5! / (2! * 3!) = 5 * 4 * 3 / (2 * 1 * 3 * 2 * 1) = 10.

Therefore, with three numbers allowing repetition and order not mattering, there are ten different combinations.

4. Combination with repetition, order matters

Lastly, if repetition is allowed within a combination and the order matters, we can use the formula for permutations with repetition:

n^r

where n is the number of distinct elements and r is the number of elements selected at a time.

In the case of three numbers, the formula would be 3^3 = 27. Thus, there are 27 different combinations when repetition is allowed and order matters.

The number of combinations that can be made with a set of three numbers varies depending on whether repetition is allowed and if the order matters. By understanding the concepts of combinations and permutations, we can determine the number of unique combinations in each scenario. Remember to consider whether repetition is allowed and if the order is significant when calculating the possible combinations of a set of three numbers.

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