A permutation (or arrangement) is an ordering of elements from a set. We will show how to count the number of permutations of any finite set of elements.
1. Lisa has 5 different ornaments she wants to put in a line on her mantle. How many ways can she arrange the ornaments?
Solution: We can think of Lisa’s Mantle as having five positions. There are five ornaments, so we have 5 choices for which ornament we put into the first position. After we place the first ornament, we will have 4 choices of which ornament to put into the second position (since one ornament is already placed). Repeating this argument, there are 3 choices for the third position, 2 choices for the fourth position and 1 choice for the last position. Since we are putting an ornament in each position, we should use the Rule of Product. So the total number of ways to place the ornaments is \( 5 \times 4 \times 3 \times 2 \times 1 = 120\).
The notation \( n!\) (where \( n\) is a positive integer) refers to the product of all positive integers from \( 1\) to \( n\). Note: \( 0!\) is often defined to be \( 1\), which is the convention for the empty product.
Now, we are ready to generalize the problem. We can replace \( 5\) with \( n\), and work through the proof again.
2. Lisa has \( n\) different ornaments she wants to put in a line on her mantle. How many ways can she arrange the ornaments?
Solution: We can think of Lisa’s Mantle as having \( n\) positions. There are \( n\) ornaments, so we have \( n\) choices for which ornament to place in the first position. After we place the first ornament, we will have \( n-1\) choices for the second position. Repeating this argument, there are \( n-2\) choices for the third position, \( n-3\) choices for the fourth position, and so on. For the \( n\)th position, there is \( n - (n-1)= 1\) choice. Now, by the Rule of Product, the total number of ways to place the ornaments is
\[ n \times (n-1) \times (n-2) \times (n-3) \times \ldots \times 1 = n!\]
3. Lisa has 4 different dog ornaments and 6 different cat ornaments that she wants to place on her mantle. All of the dog ornaments should be consecutive and the cat ornaments should also be consecutive. How many ways can they be arranged?
Solution: We have to decide if we want to place the dog ornaments first, or the cat ornaments first, which gives us 2 possibilities. We can arrange the dog ornaments in \( 6!\) ways, and the cat ornaments in \( 4!\) ways. Hence, by the Rule of product, there are \( 2 \times 6! \times 4! = 1728\) ways to arrange the ornaments.