Number of Ways to Create An Ordered List or a Group
- These are counting problems. They are used here as an intro to important ideas
in probability. They should be considered before reading the page Binomial Distribution.
- 1. A club has 4 members and wishes to elect a president, vice president, secretary, and treasurer. In how many ways can this be done?
- There are 4 ways to select a president, but then only 3 ways to choose the vp, but then only 2 ways to choose the secretary, and
then only 1 way to choose the treasurer,
- so, there are 4(3)(2)(1) or 24 ways to do this.
- Using members A, B, C, and D, here is a tree diagram and the sample space with 24 ways.
- 2. A club has n members and wishes each member to hold an elected office, beginning with the president, vice president, secretary, and treasurer. In how many ways can this be done?
- The answer is n factorial ways. There are n ways to fill the first office, n-1 ways to fill the 2nd office, n-2 ways to
fill the 3rd office, and so on. There are n! or n(n-1)(n-2)(n-3) ... (3)(2)(1) ways.
- 3. A club has 4 members and wishes to elect a president and then a vice president. In how many ways can this be done?
- There are (4)(3) or 12 ways to do this.
- Here order counts. A list is required. For a source set of n elements, if order is important and x elements are
chosen for the ordered list, a permutation of n things taken x at a time is required, Pn,x. There are n! / (n -x)! ways to do this.
| || |
| ||n!|| ||(number of ordered lists given n things)|
| ||(n-x)!|| ||(shorten the list so only x things are used)|
- In this problem, an ordered list of 4 elements, take 2 elements is required, P4,2.
There are 4! / (4-2)! = 24/2 = 12 ways to do this.
|P4,2 = ||
||= 12 |