Combinatorics Challenge Problems

Revision as of 12:00, 23 April 2020 by Shiamk (talk | contribs) (Problem 7)

Problem 1

How many distinguishable towers consisting of $8$ blocks can be built with $2$ red blocks, $4$ pink blocks, and $2$ yellow blocks?

Answer: $(420)$


Problem 2

How many ways are there to seat $6$ people around the circle if $3$ of them insist on staying together?(All people are distinct)

Answer: $(36)$


Problem 3

When $6$ fair $6$ sided dice are rolled, what is the probability that the sum of the numbers facing up top is $10$?

Answer: ($\frac{7}{2592}$)


Problem 4

How many different ways are there to buy $8$ fruits when the choices are apples, pears, and oranges?

Answer: $(45)$


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: $(\frac{25}{63})$


Problem 6*

$3$ 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: $(\frac{3}{4})$


Problem 7

A fair coin is tossed $10$ times, each toss resulting in heads or tails. What is the probability that after all $10$ tosses, that there were atleast $6$ heads?

Answer: $(\frac{385}{1024})$


Problem 8

A frog sitting at the point $(1, 2)$ begins a sequence of jumps, where each jump is parallel to one of the coordinate axes and has length $1$, 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 $(0,0), (0,4), (4,4),$ and $(4,0)$. What is the probability that the sequence of jumps ends on a vertical side of the square$?$ (Source: AMC 12A 2020).