Combinatorics · Chapter 5

Casework

Too tricky to count in one go? Split it into cases that don't overlap, count each one, and add them up.

What's this chapter about?

5.1 Casework Fundamentals
5.2 Harder Casework Examples
5.3 Practice Problems

You'll need

5.10%

Casework Fundamentals

In this section you will be able to split a counting or probability problem into cases that don't overlap and don't miss anything, then add them up.

You roll two dice. In how many ways can they add up to 3? You could list all 36 rolls. Or you could ask one question at a time: what if the first die is a 1? What if it’s a 2?

That second way is casework. Split the problem into cases, count each case, and add.

Look:good cases never overlap and never miss anything. Then the case counts add up to the answer.

Exploreplay first, formulas later

Try this: Make a case for "1st roll = 1", then "1st roll = 2". Which tiles are still dashed? Add a case that catches them.
Try this: Clear the cases and add "doubles" and "sum is odd". Why does the total come out wrong?
Try this: Press Show me on the Casework card. Which tiles turn red, and why can't you just add those bins?

Name it

Concept 5.1.1 · Casework

Casework is solving counting or probability problems by considering the different cases and adding them together. Many counting or probability problems can be solved by dividing a problem into several cases and calculating arrangements and probabilities for each case before summing them together.

You should try casework when there are no other obvious approaches, as it is one of the most common techniques on the AMC 8.

Good cases are disjoint (no outcome is in two cases) and complete (every outcome is in some case). Then the case sizes add up to the answer.

Watch itpredict, then reveal

Example 5.1

Two dice are rolled. What is the number of the ways that the sum of the numbers rolled is 3?

1
How many ways are there for two dice to sum to 3? Let's take cases on the first roll. Case 1: the first roll is a 1. What must the second roll be?

Predict first:
Your prediction

Example 5.2

Two cards are dealt from a deck of four red cards labeled A, B, C, D and four green cards labeled A, B, C, D. Winning pair is two of the same color or two of the same letter. How many ways are there to draw a winning pair?

1
What are the different ways to draw a winning pair? Can a pair be both the same color and the same letter?

Predict first:

Example 5.3

Omega Learn

Sohil has 10 boxes. Each box has a green, orange, red, and blue ball. He randomly chooses one of the boxes and then uniformly at random picks a ball from that box. He repeats this process of randomly choosing any box (allowed to choose same box again) and randomly choosing a ball in that box. What is the probability that the balls are the same color?

1
Sohil keeps the first ball he picks; it doesn't go back in its box. What's a good thing to take cases on?

Predict first:

open on YouTube

Try it

Problem 5.3.1

AMC 8

How many different combinations of $5 dollar bills and $2 dollar bills can be used to make a total of $17 dollars if the order of the bills does not matter?

Problem 5.3.4

AMC 8

Paul owes Paula 35 cents and has a pocket full of 5-cent coins, 10-cent coins, and 25-cent coins that he can use to pay her. What is the difference between the largest and the smallest number of coins he can use to pay her?

Problem 5.3.6

AMC 8

A fair coin is tossed 3 times. What is the probability there will be at least 2 consecutive heads?

Problem 5.3.3

How many 3-digit even numbers are there with distinct digits?

Make a case:

Two-digit numbers: count the numbers where even, digits differAnswer: ?

? + ? + …

Pick a rule chip to make your first case.

Warm-up: the same idea with two-digit numbers.

Contest cornerreal contest problems

Problem 5.3.2

AMC 8

The "Middle School Eight" basketball conference has 8 teams. Every season, each team plays every other conference team twice (home and away), and each team also plays 4 games against non-conference opponents. What is the total number of games in a season involving the "Middle School Eight" teams?

Problem 5.3.5

AMC 8

How many pairs of parallel edges does a cube have?

Problem 5.3.7

AMC 8

How many 4-digit positive integers have four different digits, where the leading digit is not zero, the integer is a multiple of 5, and 5 is the largest digit?

Problem 5.3.8

AMC 10

Three fair six-sided dice are rolled. What is the probability that the values shown on two of the dice sum to the value shown on the remaining die?

Problem 5.3.12

MATHCOUNTS

The Beavers, Ducks, Platypuses, and Narwhals are the only four basketball teams remaining in a single-elimination tournament. Each round consists of the teams playing in pairs with the winner of each game continuing to the next round. If the teams are randomly paired and each has an equal probability of winning any game, what is the probability that the Ducks and the Beavers will play each other in one of the two rounds? Express your answer as a common fraction.

Checkpoint for section 5.1

Get 2 of 3 right to master this section. One try each, no hints.

5.20%

Harder Casework Examples

In this section you will be able to pick one thing to split on, and use subcases when a case is still too big.

Some problems need cases inside cases. Think of a tree: the trunk splits into big branches, and a big branch can split into smaller ones. You count the leaves and add them up.

Pick one thing to split on, like the number of blue boxes or the hundreds digit. If a case is still hard, split it again.

Exploreplay first, formulas later

Try this: The tree splits on the hundreds digit. Predict each count before you press Reveal.
Try this: Open "hundreds digit 1". It splits again on the tens digit. Why is that a subcase?
Try this: Why does "hundreds digit 4 or more" give 0? Is it OK to keep an empty case?

Name it

Remark 5.2.1

For casework problems, pick one attribute to change and find the number of ways in each of the cases. In the previous problem, this was the number of blue boxes. Sometimes, the problem may require subcases: cases inside a case.

Watch itpredict, then reveal

Example 5.4

Omega Learn

There are 3 identical blue boxes, 6 identical green boxes, and 5 distinct items. Each of the items must be placed into either a green or blue box, and each box can contain a maximum of 1 item. Assuming all of the boxes are different, how many ways are there to do this?

Each tile lists the items that go in blue boxes.

1
There are 5 items to put in 5 of the 9 boxes. How can we divide this problem into cases? How many cases are there?

Predict first:
Your prediction

Example 5.5

AMC 8

A $\triangle$ or $\bigcirc$ is placed in each of the nine squares in a 3-by-3 grid. Shown below is a sample configuration with three $\triangle$ s in a line. How many configurations will have three $\triangle$ s in a line and three $\bigcirc$ s in a line?

1
Can the line of $\triangle$ s and the line of $\bigcirc$ s share a square?

Predict first:

open on YouTube

Try it

Problem 5.3.11

Omega Learn Math Contest

Given the set $\{1, 2, 3, 4, 5, 6, 7\}$ , how many non-empty subsets of this set are there such that the sum of the elements are even?

Problem 5.3.22(additional)

BMMT

At the Berkeley Sandwich Parlor, the famous BMT sandwich consists of up to five ingredients between the bread slices. These ingredients can be either bacon, mayo, or tomato, and ingredients of the same type are indistiguishable. If there must be at least one of each ingredient in the sandwich, and the order in which the ingredients are placed in the sandwich matters, how many possible ways are there to prepare a BMT sandwich?

Problem 5.3.23(additional)

Omega Learn

How many 4-digit numbers exist such that the 2nd digit is even, the 3rd digit is a multiple of 3 that is not a multiple of 2, and the 4th digit is a multiple of 4 if all of the digits are greater than 0 and different?

Contest cornerreal contest problems

Problem 5.3.9

AMC 10

How many rearrangements of $abcd$ are there in which no two adjacent letters are also adjacent letters in the alphabet? For example, no such rearrangements could include either $ab$ or $ba$ .

Problem 5.3.10

AMC 10

A license plate in a certain state consists of 4 digits, not necessarily distinct, and 2 letters, also not necessarily distinct. These six characters may appear in any order, except that the two letters must appear next to each other. How many distinct license plates are possible?

Problem 5.3.13

AMC 10

A farmer's rectangular field is partitioned into 2 by 2 grid of 4 rectangular sections as shown in the figure. In each section the farmer will plant one crop: corn, wheat, soybeans, or potatoes. The farmer does not want to grow corn and wheat in any two sections that share a border, and the farmer does not want to grow soybeans and potatoes in any two sections that share a border. Given these restrictions, in how many ways can the farmer choose crops to plant in each of the four sections of the field?

Problem 5.3.14

AMC 10

How many ways are there to place 3 indistinguishable red chips, 3 indistinguishable blue chips, and 3 indistinguishable green chips in the squares of a $3 \times 3$ grid so that no two chips of the same color are directly adjacent to each other, either vertically or horizontally?

Checkpoint for section 5.2

Get 2 of 3 right to master this section. One try each, no hints.

5.3

Practice Problems

In this section you will be able to mix casework with everything else you know on contest problems.

Time to mix it all up! Dice, digits, coins and grids. Before you count, ask: what should I split on? Every problem has three hints if you get stuck.

Want every problem from this chapter on one page? Open the problem set.

Mixed practice

Problem 5.3.20(additional)

Three unique fair tetrahedral die, whose faces are numbered 1, 2, 3, and 4 are rolled. How many ways can the sum of the numbers rolled be 7?

Problem 5.3.16(additional)

AMC 8

When three positive integers $a$ , $b$ , and $c$ are multiplied together, their product is 100. Suppose $a < b < c$ . In how many ways can the numbers be chosen?

Problem 5.3.17(additional)

AMC 8

A palindrome is a number that has the same value when read from left to right or from right to left. (For example, 12321 is a palindrome.) Let $N$ be the least three-digit integer which is not a palindrome but which is the sum of three distinct two-digit palindromes. What is the sum of the digits of $N$ ?

Problem 5.3.24(additional)

MATHCOUNTS

The number 40,231 is a five-digit positive integer that uses five consecutive digits, although not necessarily in order. How many such five-digit numbers are there?

Contest cornerreal contest problems

Problem 5.3.15(additional)

AMC 8

Abby, Bridget, and four of their classmates will be seated in two rows of three for a group picture, as shown.

\begin{matrix} X & X & X \\ X & X & X \end{matrix}

If the seating positions are assigned randomly, what is the probability that Abby and Bridget are adjacent to each other in the same row or the same column?

Problem 5.3.18(additional)

EMCC

Fred and George have a fair 8-sided die with the numbers 0,1,2,9,2,0,1,1 written on the sides. If Fred and George each roll the die once, what is the probability that Fred rolls a larger number than George?

Problem 5.3.19(additional)

AMC 12

On a standard die one of the dots is removed at random with each dot equally likely to be chosen. The die is then rolled. What is the probability that the top face has an odd number of dots?

Problem 5.3.21(additional)

EMCC

Chad and Jordan independently choose two-digit positive integers. The two numbers are then multiplied together. What is the probability that the result has a units digit of zero?

Problem 5.3.25(additional)

MATHCOUNTS

In the grid shown, how many ways are there to spell the word "QUEUE" by moving one square at a time either horizontally or vertically, and provided squares may be revisited?

Problem 5.3.26(additional)

EMCC

Jasmine rolls a fair 6-sided die, with faces labeled from 1 to 6, and a fair 20-sided die, with faces labeled from 1 to 20. What is the probability that the product of these two rolls, added to the sum of these two rolls, is a multiple of 3?