Statistics · Ch 6 — Permutations, Combinations and Binomial Expansion

Fundamental Principle of Counting

Before we can count how many ways an event can happen, we need a systematic way of counting instead of listing every possibility by hand. This is exactly the gap the Fundamental Principle of Counting fills, and it is the foundation on which both permutations and combinations are built — the Gujarat Std-11 Statistics syllabus (Business Mathematics & Statistics) opens this chapter with it for that reason.

Multiplication Principle (Principle of Counting)

If one operation can be performed in $m$ different ways, and after it has been performed in any one of these $m$ ways, a second independent operation can be performed in $n$ different ways, then the two operations together can be performed in

m \times n

ways.

This extends to any number of successive operations: if operations $O_1, O_2, \dots, O_k$ can be performed in $n_1, n_2, \dots, n_k$ ways respectively, the total number of ways of performing all of them in succession is

n_1 \times n_2 \times \cdots \times n_k

Example. A student appearing for a Gujarat board practical exam has to first pick one of 4 lab coats and then one of 3 identity badges. Total ways to pick a coat-and-badge combination $= 4 \times 3 = 12$ .

Addition Principle

If an operation can be performed in $m$ ways, and a second, mutually exclusive operation (i.e., the two cannot happen together — you do one or the other) can be performed in $n$ ways, then either operation can be performed in

m + n

ways.

Example. A student wants to travel from Ahmedabad to Vadodara either by 3 available bus routes or by 2 available train routes (not both on the same trip). The number of ways to choose the mode of travel $= 3 + 2 = 5$ .

Telling the two apart

Situation	Which principle	Operation
Do this AND then that	Multiplication	Multiply the ways
Do this OR that (mutually exclusive)	Addition	Add the ways

This distinction — AND multiplies, OR adds — is the single most useful habit to build before moving to permutations and combinations, because every permutation/combination formula is really just a repeated, compact application of the multiplication principle.

Definition 1Multiplication (Fundamental) Principle of Counting

If one event can occur in $m$ ways and, independently, a second event can occur in $n$ ways, both events together (event 1 and event 2) can occur in $m \times n$ ways.

Definition 2Addition Principle of Counting

If an event can occur in $m$ ways or a mutually exclusive second event can occur in $n$ ways, either event occurring can happen in $m + n$ ways.

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