We may also consider the possibility of arranging, in a row, r objects chosen from a set of n. We have n choices for the first space, n - 1 for the second, n - 2 for the third, and so on. Finally we have n - r + 1 choices for the rth space, giving a total of n(n - 1)(n - 2)(n - r + 1) possible arrangements (or permutations). This can be written in terms of factorials as follows:

It should be noted that this is not the number of combinations of r objects taken from a set of n, since in permutations order is important; in combinations it is not. For example, if we consider the three objects a, b, and c, the number of permutations of two objects chosen from them is  the arrangements ab, ba, bc, cb, ca, ac. However, the number of combinations is  and the combinations are a and b, b and c, and a and c.

Next we define even and odd permutations of 1, 2, ..., n; this topic is used in Chapter 9 and in higher algebra.

We begin with the case n = 3, that is the numbers 1, 2, 3. With each permutation

of these three numbers, we associate the product of differences

If p is positive, the associated permutation is called even; if p is negative, the associated permutation is odd. Three of the 3! permutations of 1, 2, 3 are even and three are odd. The even ones are listed in the first column, and the odd ones in the second column:
 
 

1, 2, 3		1, 3, 2
2, 3, 1		2, 1, 3
3, 1, 2		3, 2, 1

 

Tuesday, May 12, 1998