If we think about what happens after the first toss, we have H or T. For each of those outcomes, we get H or T on the second toss, so we now have 4 outcomes: HH, HT, TH, TT
We can also see the possibilies using a possibility tree, where the first (top) level represents the first toss, the second level, the second toss, and the third level, the third toss. The blue “paths” show the outcomes with exactly two Heads.
In Example 9.2.1, we saw that each toss doubled the number of outcomes from the previous toss. For example, if we tossed the coin 4 times, we would have \(2^4=16\) outcomes. We are using the multiplication rule to determine the number of outcomes.
First choose a letter, then another letter, then choose each of the numbers. We can use the multiplication rule since we can choose each character as a process.
A string of length \(n\) over \(s\) is a list of \(n\) characters where the characters come from set \(S\text{.}\) A binary string or bit string is a string with \(S=\{0, 1\}\text{.}\) The null string is the string of length 0, denoted \(\varepsilon\text{.}\)
An \(r\)-permutation of a set \(S\) is an ordered selection of \(r\) elements taken from the \(n\) elements of \(S\) in a row. We use the notation \(P(n, r)\) for the number of \(r\)-permutations from a set of \(n\) elements.
Let \(S=\{a, b, c, d, e\}\text{.}\) Then a 3-permutation of \(S\) is a list of 3 (non-repeating) elements from \(S\text{.}\) For example, ace, dbe, bda are all 3-permutations.
The number is a string of 4 digits. The first digit cannot be 0, so there are 9 choices: 1 through 9. The second and third digits can be anything from 0 to 9, so there are 10 choices for each. The last digit must be odd, so the only choices are 1, 3, 5, 7, 9.
The first digit cannot be 0, so there are 9 choices: 1 through 9. The second digit can be anything from 0 to 9, except for whatever the first digit is, so there are 9 choices. Similarly, there are 8 choices for the third digit, and 7 for the last digit.
A bag contains two green balls (labeled \(G_1\) and \(G_2\)) and one purple ball. A second bag contains one green ball and two purple balls (labeled \(P_1\) and \(P_2\)). Suppose the following experiment is performed: One of the two bags is chosen at random. Next a ball is randomly chosen from the bag. Then a second ball is chosen from the same bag without replacing the first ball.
A person buying a computer system is offered a choice of three models of the basic unit, two models of the keyboard, and two models of printer. How many distinct systems can be purchased?
A coin is tossed four times. Each time the result \(H\) for heads or \(T\) for tails is recorded. Assume heads and tails are equally likely on each toss.
We want to count the number of possible functions from a finite set to a finite set. You may need to recall the definition of a function, Definition 7.1.1.
