Permutations, presented in Section Subsection 1.5.1, count the number of ways to permute, or rearrange, objects. When permuting objects, some can be ignored. For example consider the alphabet \(\{a,b,c\}\text{.}\)Table 3.4.1 lists all strings of length 2 over this alphabet. Table 3.4.2 lists all permutations of two, distinct characters over this alphabet.
Sometimes the order of objects is unimportant. The number of outcomes can still be counted. These are called combinations. The combinations of two letters over this alphabet are below.
These checkpoints illustrate why the formulae, that are often memorized, work. Undestanding this allows us to perform some of the counts in the Practice section which are not simple permutation or combination formulae.
We use grouping to illustrate a relationship between permutations of part of a set, and combinations of part of a set. Perform the following tasks using the alphabet \(\{A,B,C,D,E\}\text{.}\)
If the first problem counted the number of ways to order three (3) of ten (10) people and the second problem counted the number of ways any particular three can be ordered, how can these be combined to count the number of ways to select three (3) out of ten (10) people?
