If a set contains \(N\) elements, and we want to know what are the number of ways we can order them, then there are \(N\) choices for the first element, \(N-1\) choices for the second element and so on. We can write them as \(N \times (N-1) \times (N-2) \times \dots \times 1 = N!\). Permutation tells us the number of ways we can order the set of \(N\) elements.
If there are \(N\) elements in a set and we want to know the number of subsets we can make, then we have \(2\) choices for the first element, i.e. we can either keep it in the subset or won’t keep it in the subset; \(2\) choices for the second element and so on for all the \(N\) elements. Thus we will have \(2^N\) subsets from a set containing \(N\) elements.
Example 1
What is the probability that six rolls of a six sided die all give different numbers?
Total number of possible outcomes for \(6\) rolls of a \(6\) sided dice are \(6^6\), \(6\) for the first, \(6\) for the second and so on.
Number of elements in the event: \(6\) options for the first roll, \(5\) options for the socond roll and so on. So \(6 \times 5 \times 4 \times \dots \times 1 = 6!\)
Therefore the probability of \(6\) rolls of a \(6\) sided dice all giving different outcome is: \(\frac{6!}{6^6}\)