Combinatorics Challenge Problems
Contents
Problem 1
How many distinguishable towers consisting of 
blocks can be built with 
red blocks, 
pink blocks, and yellow blocks?
Answer: 
Problem 2
How many ways are there to seat 
people around the circle if 
of them insist on staying together?(All people are distinct)
Answer: 
Problem 3
When fair sided dice are rolled, what is the probability that the sum of the numbers facing up top is 
?
Answer: (
)
Problem 4
How many different ways are there to buy fruits when the choices are apples, pears, and oranges?
Answer: 
Problem 5
Ms.Carr asks her students to read any 5 of the 10 books on a reading list. Harold randomly selects 5 books from this list, and Betty does the same. What is the probability that there are exactly 2 books that they both select? (Source: AMC 10B 2020).
Answer: 
Problem 6*
points are chosen on the circumference of a circle to form a triangle. What is the probability that the circle does not contain the center of the circle?
Answer: 
Problem 7
A fair coin is tossed times, each toss resulting in heads or tails. What is the probability that after all tosses, that there were atleast heads?
Answer: 
Problem 8
A frog sitting at the point 
begins a sequence of jumps, where each jump is parallel to one of the coordinate axes and has length 
, and the direction of each jump (up, down, right, or left) is chosen independently at random. The sequence ends when the frog reaches a side of the square with vertices 
and 
. What is the probability that the sequence of jumps ends on a vertical side of the square
(Source: AMC 12A 2020).
Answer: 
Problem 9
There are 10 people standing equally spaced around a circle. Each person knows exactly 3 of the other 9 people: the 2 people standing next to her or him, as well as the person directly across the circle. How many ways are there for the 10 people to split up into 5 pairs so that the members of each pair know each other? (Source: AMC 12B 2020).
Answer: 
