Casework: all problems

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?

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?

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

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?

How many pairs of parallel edges does a cube have?

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

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?

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?

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 $ba$.

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?

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?

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.

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?

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?

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

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?

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?

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 $N$?

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?

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?

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?

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?

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?

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?

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?

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?

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?