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Section 3.4 Combinatorics: Second Counts

Subsection 3.4.1 Terminology

Permutations, presented in Section SubsectionΒ 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 \(\{a,b,c\}\text{.}\) TableΒ 3.4.1 lists all strings of length 2 over this alphabet. TableΒ 3.4.2 lists all permutations of two, distinct characters over this alphabet.
Table 3.4.1. Strings from \(\{a,b,c\}.\)
aa ba ca
ab bb cb
ac bc cc
Table 3.4.2. Permutations of \(\{a,b,c\}.\)
ba ca
ab cb
ac bc
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.
Table 3.4.3. Permutations of \(\{a,b,c\}.\)
ab
ac bc

Subsection 3.4.2 Discover a Method

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.

Checkpoint 3.4.4.

We find a numeric relationship between the two counts and use it to conjecture an interpretation of that relationship.
(a)
How many permutations of two letters from the alphabet \(\{a,b,c\}.\) are there?
(b)
How many combinations of two letters from the alphabet \(\{a,b,c\}\) are there?

Checkpoint 3.4.5.

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 \(\{A,B,C,D,E\}\text{.}\)
(a)
List all strings of length five with no letter repeated in alphabetical order.
(b)
Group the strings such that two strings are in the same group if and only if they have the same first three letters (order matters).
(d)
Group the groups such that two groups are together if and only if the strings have the first three letters (order does not matter).

Subsection 3.4.3 Practice

Checkpoint 3.4.6.

How many permutations of three letters from the English alphabet (26 characters) are there?

Checkpoint 3.4.7.

If 10 people participate in a race and only the first three places are recorded, how many possible results are there?

Checkpoint 3.4.8.

How many permutations of the letters of the word β€˜greet’ are there?

Checkpoint 3.4.9.

(a)
Suppose 10 people apply for three, identical jobs. In how many orders can three of these 10 people be hired?
(c)
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?

Checkpoint 3.4.10.

How many ways are there to select two types of ice cream from a selection of 14?

Checkpoint 3.4.11.

If a set has 15 elements, how many subsets of 4 elements exist?