Fundamental Counting Principle

Fundamental counting principal (incorporating factorials):

Now that we are able to employ factorials to condense our calculations for the fundamental
counting principle, we can progress to more challenging examples.

Example:

A grade 12 learner has Accounting, Science, Xhosa and Mathematics textbooks on their bookshelf
and wants to rearrange them. How many different ways can the books be arranged?

4 books and 4 spaces, the number of arrangements is 4! = 4 × 3 × 2 × 1 = 24



Example:

In how many different ways can the letters of the word MARKS be arranged? Assume
that repetition of letters is not permitted.

Number of letters: 5

∴ 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑎𝑟𝑟𝑎𝑛𝑔𝑒𝑚𝑒𝑛𝑡𝑠 = 5! = 5 × 4 × 3 × 2 × 1 = 120.



Example:

4 different History books, 5 different Geography books and 2 different Science books are placed on
a book shelf.
a) how many different ways can they be arranged?
b) how many different ways can they be arranged if books of the same subject must be
placed together?

a) Total number of books is 11,

∴ 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡 𝑎𝑟𝑟𝑎𝑛𝑔𝑒𝑚𝑒𝑛𝑡𝑠 = 11! = 39916800.

b) in order to solve this problem, we can treat each subject as occupying one space:

History Geography Science
Four books, number of Five books, number of Two books, number of
arrangements 𝟒! arrangements 𝟓! arrangements 𝟐!



But since there are 3 subjects there are 3! Ways in which each of the subjects can be
arranged

∴ 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑎𝑟𝑟𝑎𝑛𝑔𝑒𝑚𝑒𝑛𝑡𝑠 = 3! × [4! × 5! × 2!] = 34560.