Choosing 2 side dishes from 5 and choosing 3 side dishes from 5 both give 10. Which identity explains that?

1. Every time r objects are chosen, the other n - r objects are simultaneously rejected. 2. So naming the chosen group and naming the rejected group carry exactly the same information. 3. The number of ways to choose r must therefore equal the number of ways to choose n - r. 4. That is the identity C(n, r) = C(n, n - r). 5. Here n = 5, so choosing 2 matches choosing 5 - 2 = 3, and both counts come out as 10. 6. Dividing P(n, r) by n does not undo the ordering, since the correction factor needed is r! rather than n. _Source: OpenStax Algebra and Trigonometry (CC BY 4.0), Ch 13 "Sequences, Probability and Counting Theory", section 13.5 Counting Principles_