Math Core

When a choice happens in stages, you don't need to list every possibility. You multiply. The multiplication principle is the engine behind nearly every counting formula, and learning to set up the stages well is the single most useful counting skill on the AMC 8.

The principle

Suppose you pick an outfit: $4$ shirts, $3$ pairs of pants, $2$ pairs of shoes. For each shirt there are $3$ pants, so $4 \cdot 3 = 12$ shirt-and-pants combinations. For each of those there are $2$ pairs of shoes, so $12 \cdot 2 = 24$ outfits.

The multiplication principle

If a task happens in stages, with $n_1$ choices for the first stage, $n_2$ choices for the second stage no matter what was chosen first, and so on, then the number of ways to do the whole task is

$n_1 \times n_2 \times n_3 \times \cdots$

The phrase "no matter what was chosen first" is the key condition. The number of choices at each stage must be the same whatever happened earlier, even if the actual options change.

Slots

For numbers, codes and words, draw one blank slot for each position and write the number of choices in each slot.

How many license plates have $2$ letters followed by $3$ digits?

$\underbrace{26}_{\text{letter}} \times \underbrace{26}_{\text{letter}} \times \underbrace{10}_{\text{digit}} \times \underbrace{10}_{\text{digit}} \times \underbrace{10}_{\text{digit}} = 676{,}000.$

How many three-digit numbers have only odd digits? Each slot has $5$ choices ( $1, 3, 5, 7, 9$ ), so $5 \cdot 5 \cdot 5 = 125$ .

Fill the most restricted slot first

When some slots have special rules, fill those first. Otherwise the number of choices for a later slot can depend on what happened earlier, and the principle breaks.

Worked example: Even numbers with distinct digits

How many even three-digit numbers have three different digits?

The units digit must be even ( $0, 2, 4, 6, 8$ ) and the hundreds digit can't be $0$ . These two rules interact, because choosing $0$ for the units digit changes how many options the hundreds digit has. Split into two cases.

Units digit $0$ : hundreds digit has $9$ choices ( $1$ to $9$ ), tens digit has $8$ left: $9 \cdot 8 = 72$ .
Units digit $2, 4, 6$ or $8$ ( $4$ choices): hundreds digit can't be $0$ or the units digit, so $8$ choices; tens digit has $8$ left (anything except those two): $4 \cdot 8 \cdot 8 = 256$ .

Total: $72 + 256 = 328$ .

Worked example: Adjacent stripes

A flag has $4$ horizontal stripes. Each is painted one of $5$ colors, and neighboring stripes must be different colors. How many flags are possible?

Top stripe: $5$ choices. Each stripe below it: any color except the one directly above it, so $4$ choices. Total: $5 \cdot 4 \cdot 4 \cdot 4 = 320$ .

Notice the colors that are allowed change from flag to flag, but the number of allowed colors is always $4$ . That's all the principle needs.

Worked example: Adding and multiplying together

Three roads join town $A$ to town $B$ , four roads join $B$ to $C$ , and two more roads go directly from $A$ to $C$ . How many routes go from $A$ to $C$ without visiting a town twice?

Split on whether the route passes through $B$ . Through $B$ : $3 \cdot 4 = 12$ . Direct: $2$ . Total: $12 + 2 = 14$ .

Use multiplication for "and then" (stages) and addition for "or" (separate cases).

Common mistake

Don't forget that a leading digit can't be $0$ . "Four-digit PIN codes" allow a leading $0$ ( $10^4$ of them); "four-digit numbers" don't ( $9 \cdot 10^3 = 9000$ ). Read carefully which one the problem means.

Practice

Practice 1

A deli offers $3$ kinds of bread, $5$ kinds of meat and $4$ kinds of cheese. A sandwich uses one of each. How many different sandwiches are possible?