If we wanted to go into more detail, we could divide step 1 into 4 smaller steps - by lining up the acrobats one at a time. Thus the number of possible pictures is the product Step 5 - position the fourth acrobat : 2 ways Step 4 - position the third acrobat : 2 ways Step 3 - position the second acrobat : 2 ways Step 2 - position the first acrobat : 2 ways Step 1 - line up the 4 acrobats : 4! = 24 ways We can divide the procedure of lining up and positioning the acrobats into five steps: This problem combines our permutation formula with the multiplication principle. Each acrobat has the option of standing on his/her feet or standing on his/her head. The number of possible permutations isįour acrobats will line up for a picture. The four steps are to give Fido to a cousin, in one of 4 ways, then Lassie to another cousin in one of 3 ways, then Rover to a cousin in one of 2 ways, and finally Spot to the last cousin. In how many ways can she distribute the four puppies among her cousins? As she has no space for more than one dog, she will give one puppy to each of her four cousins. The coach may select the leadoff batter in 9 ways, then the second batter in 8 ways, the third batter in 7 ways, etc., until the last batter can be chosen in only one way. How many ways can the coach make out the batting order? The number of ways the teacher can form 5 couples isĪ baseball team has 9 starting players. Then for woman # 2 there are only 4 choices remaining, for woman # 3 only 3 choices, for woman # 4 there are 2 choices, and finally for woman # 5 there is but 1 choice. There are 5 ways of choosing a male partner for woman # 1. We divide the procedure of forming couples into 5 steps. How many ways can the teacher form 5 couples to dance the waltz? therefore, the number of possible permutations isĪ dance class has 5 men students and 5 women students.
There are 6 choices for first, then 5 choices for second, 4 for third, etc. First we choose a first place winner, then a second place finisher, then third, fourth, fifth, and sixth. There are six steps in ordering the horses. In how many different orders can the horses complete the race? There are many other ways of ordering objects besides lining them up for a picture, as the ensuing examples will demonstrate. The number of possible permutations of N objects is N! Since obviously our method of ordering ladies should work just as well with general objects, we have the following rule: Some other factorials areĪ permutation of a group of objects is a way of placing the objects in an order. More generally, for any positive integer N the notation N! designates the product We use the shorthand notation 10! to refer to this long product. The number of permutations of ten women is Of course the same technique works with ten ladies, except that we need ten slots and ten steps instead of three. By the multiplication principle, the number of ways of completing the entire procedure by doing all three steps is Finally, after filling the first two slots, there is only one lady left, so there is just one way to fill the last slot. After this first step is complete, there are then two ways of filling the second slot, as there are only two ladies left. We have three ways to place a lady in the first slot. In arranging the ladies there are three corresponding steps we must fill the first slot, then the second slot, and then the third. Going back to the case of three ladies, we imagine that there are three slots where the ladies may stand:
The way to do this is to divide the procedure of lining up the women into steps, and then use the multiplication principle. Thus it appears that we must devise a method of counting the permutations without actually writing them all down. Possibilities referring to the ladies by the letters A, B, C, we discover theĮach one of the arrangements of the women is called a permutation altogether there are six possible permutations.īut what if there were ten ladies? Now the problem becomes more serious, as it would take forever to write down all the possibilities, and even if you tried you would never know whether you left any out. Three women, it is not difficult to ponder a moment and write down all the In how many different ways can you line up the women? As there are only Photographer you can arrange the women from left to right in the order you Of your three friends, Alice, Betty, and Cindy, lined up in a row.
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