MATC44H3 Chap.4 The Addition and Multiplication Principles
The Addition and Multiplication Principles
Two rules beneath every count
The addition principle says that when the ways of several events are pairwise disjoint, the number of ways for one of them to happen is the sum of their counts; for sets, the size of a disjoint union is the sum of the sizes.
The multiplication principle says that when an outcome is built in ordered stages, the number of outcomes is the product of the numbers of choices at each stage; for sets, the size of a Cartesian product is the product of the sizes.
Using them together
The lecture counts the positive divisors of 600 by writing each divisor as a product of the prime factors of 600 with bounded exponents, which turns the count into a product.
It then counts ordered triples whose first entry is smaller than the other two by fixing the first entry, multiplying the choices for the others, and adding over every value of the first entry, which produces a sum of squares.
The practice set counts integers with a 3 in exactly one of the last two digits by four disjoint digit patterns.
Where each rule breaks
Adding is wrong when the cases overlap, because shared outcomes are counted twice; inclusion and exclusion, scheduled for week 10, repairs that. Multiplying is wrong when the number of choices at one stage depends on earlier choices, although the choices themselves may change.
Functions, bijections and Cartesian products in the practice set are direct applications of the multiplication principle.
What to practise
Divisor counts, digit-condition counts and sums over a fixed entry are the shapes to rehearse, each written with the principle named at every step.
What this chapter covers
- 01
Addition principle for disjoint events and sets
- 02
Multiplication principle and Cartesian products
- 03
Counting divisors from prime exponents
- 04
Splitting a count by the value of one entry
- 05
Disjoint digit-pattern cases
- 06
Counting functions and bijections
Worked example · free
Divisors of 360
- 1Factor: 360 = 2³ · 3² · 5.
- 1A divisor is 2ᵃ · 3ᵇ · 5ᶜ with 0 ≤ a ≤ 3, 0 ≤ b ≤ 2 and 0 ≤ c ≤ 1.
- 2By the multiplication principle there are 4 × 3 × 2 = 24 exponent choices, hence 24 divisors.
Key terms
- Disjoint Union
- A disjoint union is a union of sets that share no elements, so its size is the sum of their sizes.
- Cartesian Product
- The Cartesian product of sets is the set of ordered tuples taking one entry from each set.
- Prime Factorization
- A prime factorization writes an integer greater than 1 as a product of primes, uniquely up to order.
- Bijection
- A bijection is a function that pairs every element of one set with exactly one element of another.
The Addition and Multiplication Principles FAQ
How do I know whether to add or multiply?
Ask what one outcome looks like. If it is a single option taken from one of several separate groups, add the group sizes. If it is a sequence of decisions, one from each stage, multiply the stage counts. Most problems multiply inside each case and then add the cases.
Why does the divisor formula add one to each exponent?
A divisor can use each prime with any exponent from zero up to the exponent in the factorization. Zero counts as a choice because a divisor need not use that prime at all. So a prime appearing with exponent a gives a + 1 choices, and the multiplication principle combines them.
What if the cases in my count overlap?
Then the addition principle does not apply, because overlapping outcomes would be counted more than once. Redesign the cases so that each outcome falls in exactly one, for instance by splitting on the position of a single special digit, or wait for inclusion and exclusion later in the term.
How many functions are there from one finite set to another?
If the domain has m elements and the codomain has n, each element of the domain independently chooses an image, so there are n to the power m functions. A bijection between two sets of size n is an ordering of the codomain, so there are n factorial of them.
Exam move
Rewrite the two principles in set language from memory. Then count three things by hand: the divisors of a number of your choice, the integers below 1,000 with a digit condition of your choice, and the ordered pairs from one to n with a strict inequality. For each, state which principle every step uses.
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