Suppose you want to distribute 8 identical mini key-lime flavored dog biscuits to your 4 identical Welsh Corgi clones, so that each Corgi gets at least one biscuit.

(a) Carefully explain why the answer is P₄(8), and find the numerical value of this answer.

(b) Compare P₄(8) to 7C3 . What related counting problem does 7C3 count? What is the numerical value of 7C3 ? Why isn't P₄(8) 4! = 7C3 (alternatively, why isn't P₄(8) = 1/4! 7C3)?

(c) Compare P₄(8) to S(8, 4). What related counting problem does S (8, 4) count? What is its numerical value? Why isn't P₄(8) 8! = S (8, 4) (alternatively, why isn't P₄(8) = 1/8! S (8, 4))?