By appealing to the meanings and not the formulas of combina

By appealing to the meanings (and not the formulas) of combinations or permutations, briefly explain why each of the following is true. Example. (n n - 1) = n means there are n ways to choose n - 1 objects out of n. We have n choices for what object to exclude from the set. (a) (n 4) = (n n - r) (b0 (n n) = 1 (c) _nP_1 = n

Solution

a) It means, the number of combinations of n different things taken r at a time is equal to the number ofcombinations of n different things taken (n-r) at a time.

every time we select a group of r things, we leave behind a group of (n-r) things. so for every combination of (n-r) things there corresponds a combination of r things.

b) It means the number of combinations of n different things taken n at a time is 1. That simply means from n number of things, number of groups of n items that can be formed is1.

c) It means the number of different arrangements of n things, taking 1 at a time is n.