Section 5.4 Summary

1. The Multiplication Rule for Counting provides the total number of different possible outcomes for a series of events.
2. A permutation ${\hspace{0.17em}}_n{P}_r$ is an arrangement in which
   • $r$ items are chosen from $n$ distinct items.
   • repetition of items is not allowed.
   • the order of the items is important.

   In a permutation, order is important. In a combination, order does not matter. A combination ${\hspace{0.17em}}_n{C}_r$ is an arrangement in which

   • $r$ items are chosen from $n$ distinct items.
   • repetition of items is not allowed.
   • the order of the items is not important.
3. Combinations may be used to calculate certain probabilities. For such problems, use the following steps:
   • Step 1 Confirm that the desired probability involves a combination.
   • Step 2 Find $N\left(E\right)$, the number of outcomes in event E.
   • Step 3 Find $N\left(S\right)$, the number of outcomes in the sample space.

   • Step 4 Assuming that each possible combination is equally likely, find the probability of event $E$ as follows:

     $P\left(E\right)={\displaystyle \frac{N\left(E\right)}{N\left(S\right)}}$