Topic 2 · Test

Unit 2 test: Counting and Probability

15 questions. Answer them all, then submit to see your score. Solutions unlock after you submit.

This mock contest covers the whole Counting and Probability unit: casework and organized lists, the multiplication principle, permutations, combinations, complementary counting, basic probability, and Venn diagrams.

Question 1

In how many ways can you make $40$ cents using nickels, dimes and quarters? (You don't have to use every kind of coin.)

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 2

How many non-congruent isosceles triangles have integer side lengths and a perimeter of $20$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 3

How many four-digit numbers have four different digits, all of them odd?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 4

How many three-digit numbers greater than $500$ have three different digits?

Question 5

How many distinct arrangements are there of the letters in $\text{GAMMA}$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 6

Seven people stand in a row. In how many arrangements do three particular friends stand next to each other?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 7

Ten points are marked on a circle. How many quadrilaterals have all four vertices among these points?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 8

A robot walks from $(0, 0)$ to $(5, 3)$ , one unit at a time, moving only right or up. How many of its possible paths pass through the point $(2, 1)$ ?

Question 9

How many three-digit numbers have at least one digit equal to $0$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 10

The letters A, B, C, D, E are arranged in a row. In how many arrangements is A not first and B not last?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 11

Two fair dice are rolled. What is the probability that the two numbers differ by exactly $2$ ? Express your answer as a common fraction.

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 12

A bag holds $5$ red and $3$ blue marbles. Three marbles are drawn at random without replacement. What is the probability that all three are the same color?

Question 13

A three-digit number is chosen at random. What is the probability that its digits are in strictly increasing order from left to right (like $258$ )? Express your answer as a common fraction.

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 14

How many integers from $1$ to $200$ are divisible by $6$ or by $8$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 15

A club of $50$ students has $28$ who play soccer, $20$ who swim and $18$ who run track. Also $10$ play soccer and swim, $8$ play soccer and run track, $6$ swim and run track, and $3$ do all three. How many students do none of the three sports?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.