In Try This 3 you may have noticed that the column and the total number of possible permutations are the same. This formula provides you with a way to calculate the permutations without the need for tree diagrams or lists. In the next scenario you will see how blanks or the formula can be used to solve a problem.

 

Consider the following scenario.

 

Six people are boarding an empty bus that has 30 seats. In how many different ways may the people be seated?

 

Solving the Problem Using Blanks

 

If you would like to solve this scenario by drawing blanks, draw your blanks for the elements that are most restricted. In this case there are fewer people on the bus than there are seats, so draw the blanks to represent each person. Then fill in the blanks with the number of seats the people may choose from.

 

 

Since the first person who gets on the bus has 30 seats to choose from, 30 is placed in the first blank. Now one seat is taken, so the second person has 29 seats to choose from, and so on.

 

When you  complete this solution by filling in all six blanks, you get 30 × 29 × 28 × 27 × 26 × 25, or 427 518 000 ways.

 

Solving the Problem Using the Permutations Formula

 

In this example, n is 30 and represents the total number of seats; r is 6 and represents the number of people who choose a seat. You are finding the number of ways of choosing 6 items (r) from 30 items (n), which can be represented by

 

 

When you are presented with permutation questions, it is important to be able to identify n (the total number of objects there are to select from) and r (the number of objects being selected each time). Some questions will require you to do a little word deciphering to find these values. You can practise this deciphering as you solve the Self-Check problems.

 

Self-Check 1

1. In how many ways can the president, vice-president, and secretary of a student council be selected from 30 people? Leave your answer in factorial form. Answer
2. How many four-letter permutations can be formed from the word TRAVEL? State your answer in factorial notation and then evaluate the number of four-letter permutations. Answer
   
3. 
   Consider the National Hockey League (NHL) playoffs. There are 15 teams in the Western Conference of the NHL. In how many ways can the conference quarterfinals be matched up? Keep in mind that in the quarterfinals the first-place team plays the eighth-place squad, second place plays seventh place, third place plays sixth place, and, finally, the fourth-place team plays the fifth-place finisher.
   1. In how many ways can the quarterfinals be played? Answer
   2. Does it matter in what order you place the teams? Do you have to fill in the first-place team first, the second-place team second, and so on? Answer
   3. Can you represent your solution using factorial notation? Answer