The number of combinations of choosing 5 numbers out of 10 is 252 if the order doesn't matter. This is according to the reference, which states, "If you want permutations (order is important), that's 252 possible values."
However, the reference seems to confuse "combinations" with "permutations". It says "permutations (order is important)" but then states the value 252, which is actually the number of combinations where order is not important.
Here's a breakdown:
- Combinations: When the order of selection doesn't matter. The formula for combinations is nCr = n! / (r! * (n-r)!), where n is the total number of items, and r is the number of items being chosen. In this case, n=10 and r=5.
- Permutations: When the order of selection does matter. The formula for permutations is nPr = n! / (n-r)!. In this case, n=10 and r=5.
So, we have two possible interpretations, depending on whether order matters or not.
Case 1: Order Doesn't Matter (Combinations)
If the order of the 5 numbers does not matter, we are looking for combinations. Using the formula:
10C5 = 10! / (5! 5!) = (10 9 8 7 6) / (5 4 3 2 * 1) = 252
Therefore, there are 252 combinations of 5 numbers out of 10.
Case 2: Order Matters (Permutations)
If the order of the 5 numbers does matter, we are looking for permutations. Using the formula:
10P5 = 10! / (10-5)! = 10! / 5! = 10 9 8 7 6 = 30240
Therefore, there are 30,240 permutations of 5 numbers out of 10.
Given the original question asks for "combinations," the most likely answer is:
There are 252 combinations of 5 numbers out of 10.
