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        We can think of a 1-1 correspondence as “an ordering of the selections” from two groups of objects. In an application we may wonder “How many different ways assignments can be made?” or “How many different ways objects can be ranked in an order?”

         Example:  Cary, Dana, and Pat are elected to be president, secretary, and treasurer. What are the possible office assignments? How many assignments are possible? We could elect Cary as president, Dana as secretary, and Pat as treasure, but there are other possibilities.

         Solution:  We have two sets: the three people and the three offices. The assignment of each of the people to a particular office where each person can hold only one office is a 1-1 correspondence. The number of possibilities is the number of ways a 1-1 correspondence can be formed. Here is an organized way to list them:

  Cary – President

Dana – Secretary

    Pat – Treasurer 

Cary – Secretary

Dana – President

Pat – Treasurer 

Cary – Treasurer

Dana – President

  Pat – Secretary 

 
Cary – President

 Dana – Treasurer

   Pat – Secretary


Cary – Secretary

Dana – Treasurer

   Pat – President


Cary – Treasurer

Dana – Secretary

  Pat – President

 

       We have six ways of making the assignments. Further, note that 6 = 3 · 2 · 1. This result follows from the fact that we have three choices for Cary, once that choice was made there are two choices for Dana, and once that choice is made there is only one choice remaining for Pat. This idea of multiplying the number of choices for each to find the total number of possibilities will be covered later in the course in Session 7 with the Fundamental Counting Principle.

       Here is another way to illustrate the solution by just using the first letter of each name:
 We have two sets: people A = {C, D, P} and offices B = {p, s, t}. Here is an organized way to list the six possible office assignments: 

 C    D   P

 |     |    |

  p    s    t  

 C   D  P

 |    |    |

  p    t     s  

 C   D   P

 |     |    |

  s    p    t  

C   D   P

 |     |    |

  s    t    p  

C   D   P

 |     |    |

  t    p    s  

C   D   P

 |     |    |

  t    s    p