Training Camp (cont 5)

Example 4 explores how to use to solve a question if some of the objects in a group are identical.

How many numbers can you produce by arranging the digits 1, 2, 2, 2, 5, 5, 7, 8, 9 if the digits cannot be repeated?   There are nine digits, but some are identical. The question, then, is How many ways can you arrange nine non-distinct objects? Use the formula: In this example, there are 30 240 arrangements of the nine digits.

 

Notice that this answer is smaller than the answer in Example 3. When objects are identical, arrangements such as 12 2 5 2 5789 and 12 2 5 2 5789 are considered the same and are counted only once. However, in Example 3 the objects are distinct; so, no arrangements are left uncounted. A larger number of arrangements are inevitable in this situation.

In the previous question, the words if the digits cannot be repeated can be confusing. You may think that because the digit 2 appears three times in the list and the digit 5 appears twice, they are being repeated. This is not the case. Cannot be repeated means that after a digit is used in the arrangement it cannot be used again. Therefore, 1 can appear in the arrangement only one time; each 2 can appear in the arrangement only one time; each 5 can appear in the arrangement only one time and so on. The number 111125557 is not acceptable because 1 is being repeated and one of the 5's is being repeated.