Assuming you are using "combinations" in the mathematical sense where order doesn't matter (if order does matter it would be "permutations"), there are 22C5 = 22!/5!17! = 26,334 possible combinations of 5 numbers from 22.
They start {1, 2, 3, 4, 5}, {1, 2, 3, 4, 6}, {1, 2, 3, 4, 7}, ... and end ... {16, 19, 20, 21, 22}, {17, 19, 20, 21, 22}, {18, 19, 20, 21, 22}; I'll leave the 26,328 combinations in the middle for you to list.
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There are 23C3 = 23!/(20!*3!) = 23*22*21/(3*2*1) = 1,771 combinations.
If each number is different from all of the other numbers and the combinations are considered distinct unless they contain the same numbers in the same order, the answer is 25 X 24 X 23 X 22 X 21 = 6,375,600. If the combinations are distinct only if they do not contain the same numbers, irrespective of order, then the answer is (25 X 24 X 23 X 22 X 21)(5 x 4 x 3 x 2) = 53,130.
There are many combinations of five numbers that can add up to 22. One example is 1, 2, 3, 4, and 12. Another combination could be 0, 5, 5, 6, and 6. The specific numbers can vary widely as long as their sum equals 22.
If the numbers are all different, then 25C5 = 25*24*23*22*21/(5*4*3*2*1) = 53,130.
1 and 11 and 21 and 31 and 42 and 12 and 22 and 32 and 43 and 13 and 23 and 33 and 44 and 14 and 24 and 34 and 416 combinations