Combinatorics
Lily wants to buy four donuts from an ample supply of three types of donuts: glazed, chocolate, and powdered. This many different selections are possible.
What is 15? (2001 AMC 10 #19)
There are this many distinguishable arrangements of 1 brown tile, 1 purple tile, 2 green tiles, and 3 yellow tiles in a row from left to right. (Tiles of the same color are indistinguishable.)
What is 420? (2020 AMC 10B #5)
Henry has a 3 foot by 2 foot pan of cornbread, which is cut into pieces that measure 2 inches by 2 inches. This is the amount of pieces the pan contains.
Ella drove 96 miles in 90 minutes. Her average speed during the first 30 minutes was 60mph, and her average speed during the second 30 minutes was 65mph. This is her average speed, in mph, of her last 30 minutes
What is 67? (2018 AMC 10B #2)
This is the hundreds digit of (20!-15!)
What is 0? (2019 AMC 10A #2)
Kaden is to select six cookies from a tray containing only chocolate chip, oatmeal, and peanut butter cookies. There are at least six of each of these three kinds of cookies on the tray. This many different assortments of six cookies can be selected.
What is 28? (2003 AMC 10A #21)
This many positive cubes divide 3!*5!*7!
A three dimensional rectangular box has dimension X by Y by Z, and faces with areas of 24, 24, 48, 48, 72, and 72 square units. This is X + Y + Z
What is 22? (2018 AMC 10B #4)
The product of three integers is 60. What is the least possible positive sum of the three integers?
What is 3? (2024 AMC 10A #7)
A unit of blood expires after 10! seconds. Akshath donates a unit of blood on January 1 at noon. This is the day the blood expires (There are 31 days in January)
When is February 12? (2018 AMC 10A #3)
When 7 fair standard 6-sided dice are thrown, the probability that the sum of the numbers on the top faces is 10 can be written as n/(67).
What is 84? (2018 AMC 10A #11)
On Monday, 6 students went to the tutoring center at the same time, and each one was randomly assigned to one of the 6 tutors on duty. On Tuesday, the same 6 students showed up, the same 6 tutors were on duty, and the students were again randomly assigned to the tutors. This is the probability that exactly 2 students met with the same tutor both Monday and Tuesday.
What is 3/16? (2025 AMC 10B #11)
This is the degree measure of the acute angle formed by lines with slopes 2 and 1/3.
What is 45 degrees? (2023 AMC 12A #11)
If a+1 = b+2 = c+3 = d+4 = a+b+c+d+5, this is a+b+c+d.
What is -10/3? (2002 AMC 10A #16)
Each piece of candy costs a whole number of cents, and Avantika has enough money for either 12 pieces of red candy, 14 pieces of green candy, 15 pieces of blue candy, or n pieces of purple candy. A piece of purple candy costs 20 cents. This is the smallest possible value of n
What is 21? (2019 AMC 10b #7)
Each of 6 balls is randomly and independently painted either black or white with equal probability. This is the probability that every ball is different in color from more than half of the other 5 balls.
What is 5/16? (2021 Fall AMC 10A #13)
Ten chairs are evenly spaced around a round table and numbered clockwise from 1 through 10. Five married couples are to sit in the chairs with men and women alternating, and no one is to sit either next to or across from his/her spouse. How many seating arrangements are possible?
What is 480? (2008 AMC 10B #21)
Let ABC be an isosceles triangle with BC=AC and angle ACB = 40. Construct the circle with diameter BC, and let D and E be the other intersection points of the circle with the sides AC and AB, respectively. Let F be the intersection of the diagonals of the quadrilateral BCDE. This is the degree measure of angle BFC.
What is 110 degrees? (2019 AMC 10A #13)
Both roots of the quadratic equation x2 - 63x + k = 0 are prime numbers. This is the number of possible values of k.
What is 1? (2002 AMC 10A #14)
Clement has 2 jars that hold 95 green marbles in total, and some amount of blue marbles. Each jar has the same amount of marbles. In Jar 1, the ratio of blue:green is 9:1, and the ratio of blue:green in Jar 2 is 8:1. This is how many more blue marbles are in Jar 1 than in Jar 2
What is 5? (2019 AMC 10B #11)
A parking lot has 16 spaces in a row. Twelve cars arrive, each of which requires one parking space, and their drivers choose spaces at random from among the available spaces. Grant then arrives in his SUV, which requires 2 adjacent spaces. This is the probability that he is able to park.
What is 17/28? (2008 AMC 12B #22)
In this many ways the sequence 1,2,3,4,5 can be rearranged so that no three consecutive terms are increasing and no three consecutive terms are decreasing.
What is 32? (2021 AMC 10A #20)
Circles A, B and C are externally tangent to each other, and internally tangent to circle D. Circles B and C are congruent. Circle A has radius 1 and passes through the center of D. This is the radius of circle B.
What is 8/9? (2004 AMC 12A #23)
All the roots of the polynomial z6 - 10z5 + Az4 + Bz3 + Cz2 + Dz + 16 are positive integers, possibly repeated. This is the value of B.
What is -88? (2021 AMC 12A #12)
This is the greatest possible sum of the digits in the base-seven representation of a positive integer less than 2019.
What is 22? (2019 AMC 10B #12)