Chapter 2 Basic Counting 2.1 Product Rule: If S = S_{1} × S_{2}, then |S| = |S_{1}||S_{2}|. x = (a, b) ∈ S implies a ∈ S_{1} AND b ∈ S_{2}, then |S| = |S_{1}||S_{2}|. How many sequences consisting of one letter followed by one single digit number (0 - 9) are possible? How many different license plates are possible if 3 letters followed by 3 numbers are used? How many different DNA sequences of length 2?

2.6 Subsets 2 2.2 Sum Rule: If S = S_{1} ∪ S_{2} and S_{1} ∩ S_{2} = ∅, then |S| = |S_{1}| + |S_{2}|.

If S_{1} ∩ S_{2} = ∅ and if x ∈ S implies x ∈ S_{1} OR x ∈ S_{2}, then |S| = |S_{1}| + |S_{2}|.

How many symbols

Example 2: How many 10-digit telephone numbers are there if 1.) The area code cannot begin with a 0 or 1 and must have a 0 or 1 in the middle. 2.) Neither of the first two digits of the last 7 digits can be 0 or 1.

Example 3: How many even numbers between 100 and 1000 have distinct digits.

Example 4: How many different seven-digit numbers can be constructed out of the digits 2, 4, 8, 8, 8, 8, 8?

Example 5: How many different seven-digit numbers can be constructed out of the digits 2, 2, 8, 8, 8, 8, 8?

2.3, 2.5 Permutations and r-permuations: Suppose |S| = n. An r-permutation of S is an ordered arrangement of r of the n elements of S.

If r = n, then an r-permutation of S is a permutation of S.

P (n, r) = number of r-permutation of S where |S| = n. If r > n, then P (n, r) = P (n, 1) = n! = n(n − 1)(n − 2)...(2)(1) 0! = 1 Thm 3.2.1: If r ≤ n, then P (n, r) = \frac{n!}{(n−r)!} 2.7 r-Combinations

( \begin{matrix}n \\ r \end{matrix} ) = \frac{n!}{r!(n−r)!}

The number of f