1. Rewrite the following expressions without using P or Q.
2. Rewrite the following expressions using P or Q notation.
(a) x(x − 1)(x − 4)(x − 9)(x − 16)···(x − 900)
3. For each integer n ≥ 1, let tn be the number of strings of n letters that can be produced by concatenating (running together) copies of the strings “a”, “bb”, and “cc”.
For example, t1 = 1 (“a” is the only possible string) and t2 = 3 (“aa”, “bb” and “cc” are the possible strings).
(a) Find t3 and t4.
(b) Find a recurrence for tn that holds for all n ≥ 3. Explain why your recurrence gives tn.
4. Draw simple graphs with the following properties or explain why they do not exist.
(a) The list of vertices is: P, Q, R, S, T and the list of edges is PQ, PS, QR, QS, RT.
(b) The graph has 11 vertices and 56 edges.
(c) The graph has 8 vertices and 7 edges and is connected .
(d) The graph has 7 vertices and 11 edges and its vertices can be divided into two sets in such a way that every edge joins a vertex in one set to a vertex in the other.
