Permutations and Factorials Mathematicians call an ordering of a set of distinct o...

Permutations and Factorials

Mathematicians call an ordering of a set of distinct objects a permutation. The number of possible permutations for a set of n objects is calculated by first noticing that n choices are possible for the first position in the ordering. After that first object has been placed, note that (n - 1) choices are possible for the second position in the ordering. Next, since 2 objects have already been placed, (n - 2) choices are possible for the third position, and so on until only 1 object remains, and it must be in the last position. Altogether we have n × (n - 1) × (n - 2) × . . . × 2 × 1 permutations, or possible ways to order n objects.

The number n × (n - 1) × (n - 2) × . . . × 2 × 1 is called n factorial and is written n!. We can put the three voters in order in 3! = 3 × 2 × 1 = 6 possible ways.

Calculate each of the following numbers by expanding each numerator and denominator using the definition of a factorial and then canceling common factors.