## Friday, January 11, 2013

In a beginning statistics class, topics students are taught are basic counting principles, ways to order a certain number of objects and how many ways to select a certain number of objects from a group of objects.

A permutation is an ordered arrangement that each item in the arrangement is used once and the order of the items in the arrangement makes a difference. When we think about combinations , we consider that items are selected from the same group and no item may be used more than once. This is the same as in permutations. What makes combinations and permutations different is that the order does not matter in combinations.

How does one know whether to use combinations or permutations? The following examples will erase any confusion on this topic.

1. A local restaurant offers pizza with up to 20 toppings. One special they offer is any large pizza with 3 toppings for \$10. How many different pizzas are possible?

Solution:
The order in which the toppings are put on the pizza does not matter. A pizza with pepperoni, sausage and green peppers is the same as one with green peppers, pepperoni and sausage. Therefore, this is a problem involving combinations.

2. Ten lodge members are running for president and vice president. When the votes are counted, the member with the second highest number of votes is named vice president and the member with the highest number of votes is elected president. How many possible outcomes are possible?

Solution:
Lodge members are choosing a president and a vice president from ten members. The order in which they are chosen matters since the positions are different. Since order matters, this is a problem involving permutations.

3. A football team of 52 players wishes to elect a group of 4 players to be team captains for the season. How many different groups of 4 players can be chosen to be team captains?

Solution:
The order in which the 4 players are chosen to be team captains does not matter since they are not occupying different positions. Each player in the group is considered a captain and is equal to the other captains. Therefore, since order does not matter, this is a problem involving combinations.

4. How many different 7 digit passwords are possible from the numbers 1, 2, 3, 4, 5, 6 and 7 if no digits may be used more than once.

Solution:
The order the digits appear in the password is important. A password of 1, 3, 5, 2, 4, 6, 7 is different from 1, 2, 3, 4, 5, 6, 7. Since order matters, this is a problem involving permutations.
We use the notation n Pr to denote the number of permutations of n objects taken r at a time. For formula for permutations is as follows

n Pr = n! /( n - r)!
The number of combinations of n objects taken r at a time is given by

nCr = n!/ [r!(n - r)!]

Note that n! = n(n - 1)(n - 2).....1

This guide should clear any issues pertaining to what combinations and permutations are and when to use them.