Combinatorics Challenge Problems

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

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