How can you prove the combination formula?

GENERIC:

• Let C(n,r) be the number of ways to generate unordered combinations
• The number of ordered combinations (i.e. r-permutations) is P(n,r)
• The number of ways to order a single one of those r-permutations P(r,r)
• The total number of unordered combinations is the total number of ordered combinations (i.e. r-permutations) divided by the number of ways to order each combination
• Thus, C(n,r) = P(n,r)/P(r,r) = [n!/(n-r)!]/r!/(r-r)!] = n!/r!(n(n-r)!

SPECIFIC:

• Let C(52,5) be the number of ways to generate unordered Poker hands
• The number of ordered poker hands is P(52,5) = 311,875,200
• The number of ways to order a single poker hand is P(5,5) = 5! = 120
• The total number of unordered poker hands is the total number of ordered hands divided by the number of ways to order each hand
• Thus, C(52,5) = P(52,5)/P(5,5)