Reference no: EM132293965

1. Six persons a, b, c, d, e, and f want to sit at a dining table with six chairs.

Answer the following questions using the Product Rule.

(a) In how many different ways can they sit at the table?

(b) In how many different ways can they sit at the table such that persons e and f sit on two adjacent chairs?

(c) In how many different ways can they sit at the table such that persons e and f don't sit on the adjacent chairs?

2. Let n be a randomly chosen 7-digit positive integer.

(a) What is the probability that n is even?

(b) What is the probability that n ≥ 7, 000, 000?

(c) What is the probability that 1, 500, 000 ≤ n ≤ 6, 200, 000?

(d) What is the probability that n does not contain repeated digits and is not made of digits 0 and 5?

(e) What is the probability that n has no repeated digits and is divisible by 5?

3. In a set of 150 integers, 120 of them are divisible by 3, 60 of them are divisible by 5, and 40 of them are divisible by 15. How many integers are divisible by neither 3 nor 5 (Hint: An integer is divisible by 15 if it is divisible by both 3 and 5).

4. Let G = (V, E) be an undirected graph such that V = {v1, v2, v3, v4} and E = {e1, e2, e3}. If e1 is incident on v1 and v4, e2 connects v3 and v4, and v2 is the only endpoint of e3, answer the following question:

(a) Draw the graphical representation of graph G.

(b) Find three different walks from v1 to v3 and specify whether each one of them is a trail and/or path from v1 to v3.

(c) Construct the matrix representation of G.

(d) Draw three different subgraphs of G.

(e) Find all of the connected components of G. Is G connected?

5. Find 7 non-isomorphic graphs with three vertices and three edges.

6. Prove that the complete bipartite graph K4,6 has an Euler circuit.