Mathematics and Statistics · Ch 15 — Permutations and Combinations

The Fundamental Principle of Counting

This chapter of the Maharashtra Std XI (FYJC) commerce Mathematics and Statistics course is about counting without listing — finding how many arrangements or selections are possible when writing every one out by hand would be hopeless. These are the same standard counting principles used across the national mathematics curriculum, and they lead directly into the study of probability in the next chapter.

Everything rests on one idea, stated in two parts.

Note

Multiplication Principle (the "and" rule)

If one task can be done in $m$ ways, and after it a second, independent task can be done in $n$ ways, then the two tasks in succession can be done in $m \times n$ ways. This extends to any number of tasks: multiply the number of ways for each stage.

The multiplication principle applies when a job is completed only after all the stages are done (stage 1 and stage 2 and ...).

Example: a lunch menu offers $3$ starters and $4$ main courses. A meal of one starter and one main can be chosen in $3 \times 4 = 12$ ways.

Note

Addition Principle (the "or" rule)

If a task can be done in $m$ ways or (by a completely separate method that shares no outcome with the first) in $n$ ways, then the number of ways to do the task is $m + n$ .

The addition principle applies when a job is completed by choosing exactly one of several mutually exclusive options (method A or method B).

Example: to travel from town P to town Q there are $2$ bus routes or $3$ train routes; the journey can be made in $2 + 3 = 5$ ways.

The whole chapter is really these two rules applied with care: decide whether stages happen together (multiply) or as alternatives (add).

Definition 1Multiplication principle

If a job is done in successive independent stages with $m, n, \ldots$ ways at each stage, the whole job can be done in $m \times n \times \cdots$ ways (used for tasks joined by "and").

Definition 2Addition principle

If a job can be done by one of several mutually exclusive methods with $m, n, \ldots$ ways, the total number of ways is $m + n + \cdots$ (used for tasks joined by "or").

← Previous chapterCh 14 — Correlation Next · 2 →Factorial Notation