Answer the following questions using the Generalized Product Rule.
(a) Next week, I’m going to get really fit! On day 1, I’ll exercise for 5 minutes.
On each subsequent day, I’ll exercise 0, 1, 2, or 3 minutes more than the previous day. For example, the number of minutes that I exercise on the seven days of next week might be 5, 6, 9, 9, 9, 11, 12. How many such sequences are possible?
(b) An r-permutation of a set is a sequence of r distinct elements of that set. For example, here are all the 2-permutations of {a; b; c; d}:
How many r-permutations of an n-element set are there? Express your answer using factorial notation.
(c) How many n×n matrices are there with distinct entries drawn from {1; : : : ; p},
where p ≥n2?