Multiplication Rule for Counting

For a sequence of two events in which the first can occur in \(n\) ways and the second in \(m\) ways, the two events can occur a total of \(n\times m\) ways.


Example 1 One red and one blue chip are to be selected from a box of 12 red and 8 blue chips How many different choices are possible?



Example 2 A vehicle tag number consists of three letters followed by four digits. How many unique tag numbers can be formed?



Example 3 If 30 people are in a room, what is the probability that at least two of them were born on the same day of the year?




Combination

Combination: A collection, regardless of order, in which \(r\) objects are chosen from \(n\) different objects (\(r\le n\)) and repetition is not allowed. The total number of possible combinations is denoted by \({_n}C_{r}\), \(C(n, r)\), or \(n \choose r\).

\[{n\choose r} = \dfrac{n!}{r!(n-r)!}\]

where \(n!\) is the factorial of \(n\) such that \(n!=1\times 2\times 3 \times \cdots \times n\).

choose(n, r)

Example 4

(a). How many different groups of two items can be selected from a set of three, regardless of order?

(b). How many different groups of three items can be selected from a set of five, regardless of order?




Example 5 If five students in our class are to be randomly selected by the General Education Committee for the purpose of assessing student learning outcomes, How many different groups of five can be selected?




Permutation

Permutation: An ordered arrangement in which \(r\) objects are chosen from \(n\) different objects (\(r\le n\)) and repetition is not allowed. The total number of possible permutations is denoted by \({_n}P_{r}\), or \(P(n, r)\).

\[P(n, r) = \dfrac{n!}{(n-r)!}\]

factorial(n) / factorial(n - r)

Did you notice the relation between combination and permutation? \[P(n, r) = C(n, r)\cdot r!\] Therefore, we can also calculate permutation \(P(n, r)\) by

choose(n, r) * factorial(r)

Example 6 The England Premier League (soccer) has 20 teams. Every season, each team has two games (home and away) with every other team. How many games are there in the league every season?




Example 7 Four students \(A, B, C\), and \(D\) are scheduled to present in the same paper session on BEAR Day. What is the probability that the order of the speakers is \(ADBC\)?




Lottery

Example 8 The following information is available on the Illinois Lottery’s website. Can you explain the winning chances?




Example 9 The following information is available on the Georgia Powerball website. Can you explain the winning chances? (How to play: Choose 5 numbers from 1 to 69 and a Powerball number from 1 to 26. )

Match Prize Odds
\(+\) Jackpot 1 in \(292,201,338\)
$\(1,000,000\) 1 in \(11,688,054\)
\(+\) $\(50,000\)
$\(100\)
\(+\) $\(100\)
$\(7\)
\(+\) $\(7\)
\(+\) $\(4\)
$\(4\)