How many 5-number combinations can you make from 9 numbers?

Answer 1

-4

Answer 2

There are a couple of basic answers to the number of 5-number combinations that can be made from 9 numbers in this exercise in permutations. One answer allows the repeated use of the numbers, and the other answer does not allow for repeated use of the numbers.

If we use the numbers 1 through 9 (no zero), and we allow the repeated use of a number, we'll be drawing from what is called a replacement set. We'll see 11111, 11112, 11113, ..., 99999. There will be 9 x 9 x 9 x 9 x 9 combinations or 59,049 possible combinations. If no repeated use of a number is allowed, we'll be drawing from a non-replacement set. We'll see 12345, 12346, 12347,..., 56789. There will be 9 x 8 x 7 x 6 x 5 combinations or 15,120 possible combinations.

You can see how we did this by looking at the number of elements in a given combination, and the number of elements in the set from which a given element is drawn.

In a replacement set, we just take the number of elements in the replacement set and raise it to the power of the number of digits in the combination. In the first case, we had 9 elements and a combination of 5 elements. Following our idea, we'd have 95 combinations, which is what we said when we wrote 9 x 9 x 9 x 9 x 9 = 59,049 possible combinations.

In using a non-replacement set, we'd have one less number to draw from when chosing the next element for our combination. That's why we have one less number when we set up the permutation and write 9 x 8 x 7 x 6 x 5 to get 15,120 possible combinations. There may also be other restrictions we place on the combinations that cause us further modification of our formulation to discover how many different combinations are possible when chosing elements from a given set.