3-5-26 Section 5-5 and 5-7 Demystifying Probability: A Guide to Counting Techniques

If a task involves a sequence of choices where there are $p$ selections for the first choice, $q$ selections for the second choice, and $r$ selections for the third, the entire task can be done in $p \cdot q \cdot r \dots$ different ways.
This Multiplication Rule of Counting explains why a restaurant offering 2 appetizers, 4 entrées, and 2 desserts results in 16 total possible meal orders.

When you need to choose items from a group without repetition, you have to decide if the order of your selection is important.

A permutation is an ordered arrangement of $r$ objects chosen from $n$ distinct objects where repetition is not allowed.
Permutations are calculated using the formula $_nP_r = \frac{n!}{(n-r)!}$.
Finding out the number of ways distinct horses can cross the finish line in first, second, and third place requires a permutation because the finishing order is important.
A combination is a collection of $r$ objects chosen from $n$ distinct objects without repetition, where the order does not matter.
Combinations are calculated using the formula $_nC_r = \frac{n!}{r!(n-r)!}$.
Selecting a two-person team from a group of four friends to play golf is a combination because the order in which the two friends are selected does not matter.

If you are arranging $n$ objects where some items are identical (like $n_1$ of one kind and $n_2$ of a second kind), you cannot use the standard permutation formula.
Instead, the number of permutations is found using the formula $\frac{n!}{n_1! \cdot n_2! [cite_start]\dots n_k!}$.
This formula is necessary for problems like finding the number of distinguishable DNA sequences you can form using a specific mix of letters representing nucleotide bases.

Determining the correct probability rule or counting technique is often the hardest part of solving a problem.

Probability Rules

To find the probability of a single event with equally likely outcomes, use the classical approach formula: $P(E) = \frac{N(E)}{N(S)}$.
If the events involve the word "AND", use the General Multiplication Rule, determining first whether the events are independent.
If the events involve the word "OR", use the General Addition Rule, determining first whether the events are disjoint or mutually exclusive.

Counting Techniques Flowchart

First, ask yourself if you are making a sequence of choices.
If you are making a sequence of choices that are independent of previous stages, rely on the Multiplication Rule of Counting.
If you are not making a sequence of choices, ask yourself if the order of arrangements matters.
If order matters, and you are selecting $r$ objects from $n$ distinct objects, apply the permutation formula.
If order does not matter, apply the combination formula.