NRP Math Challenge Club
EXAMPLES
1. There are five different pairs of socks in a dark drawer. Henry picks them out randomly. What is
the minimum amount he needs to take out to be certain he has a matching pair?
a. Henry has 6 blue socks, 5 red socks, and 4 yellow socks in the drawer. What is the
minimum amount he needs to take out to be certain he has two pairs of the same colour?
What if the two pairs have to be red?
2. Erik has five different T-shirts and four pants. How many different outfits can he make?
a. Erik wants to add one of two hats to his outfit. Now how many different outfits can he
make?
b. Erik received seven pairs of sandals and two pairs of dress shoes on his birthday. Now how
many different outfits can he make?
3. Today there are twenty people in the math club.
a. We have to pick one person to be the president and another person to be the vicepresident. In how many ways can we do this?
b. What if we had to pick two people to become cheerleaders, is the answer the same?
4. 3 people enter a room and all pairs of people shake hands with each other. How many handshakes
are there?
a. What if there are ten people?
b. Instead of shaking hands, the ten people are playing a game called “Atom” and they have to
make groups of three, an “atom”. All possible atoms are made. How many groups were
made?
NRP Math Challenge Club
5. Kiernan rolls a die. What is the probability he rolled a prime number?
a. Kiernan rolls two die. What sum is the most likely? What is the probability of rolling that
sum?
b. Kiernan rolls three die. What is the probability that he rolls 16 or more?
c. What sum is the most likely when you roll three die?
6. Michelle flips two coins. What is the probability that they are the same?
a. Michelle flips five coins. What is the probability there are exactly two heads?
b. What is the probability there are at least four heads?
c. What is the probability there is at least 1 head?
d. Michelle flips three coins, what is the probability there are at least two coins that are the
same?
7. Brian has a shuffled deck of cards. What is the probability that the top two cards make a pair?
12
11
(Write the final equation, you do not need to calculate it. For example: 14 × 13 is acceptable.)
a. Brian deals the first three cards onto the table. What is the probability that they are all the
same suit?
b. Brian deals the first five cards onto the table. What is the probability that they make a
“straight”? Five cards make a straight if the ranks of the cards are five consecutive values
in: 2,3,4,5,6,7,8,9,10,J,Q,K,A. Note that K,A,2,3,4 is not a flush.
8. A group of five people wants to sit down in line and take a picture. How many different possible
pictures can they take?
a. Sharon and Dasha are two people in this group, who are best friends so they must sit
beside each other. Now how many pictures can they take?
b. Jenny and Chae-won got into a big fight, so they must not sit beside each other or they will
fight more. How many pictures can they take?
NRP Math Challenge Club
1. If I flip two pennies. What is the probability that I get at least 1 head?
2. Norma Rose Math Club needs to choose 4 people to represent them at the club fair. If there are 10
people in math club, how many ways are there to choose 4 people?
3. In a round-robin style mental math competition, each of ten contestants must match up against each of
his/her opponents exactly three times. How many matches are played in all?
4. Thirteen cards are stacked in increasing order: Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K. If you were to choose
a random card from the stack, then take that card and all cards above it and put them in the same
order at the bottom of the stack, what is the probability that the top three cards in the new stack
would still be in increasing order? For example, if you chose the 5, your new stack would be: 5, 6, 7, 8,
9, 10, J, Q, K, Ace, 2, 3, 4. Express your answer as a reduced fraction.
5. How many different arrangements are there of the word RANDOM if neither the starting nor ending
letter is a consonant (not a vowel)?
6. If the letters in the word “BINGO” are arranged at random, what is the probability that the two vowels
will not be immediately next to each other in the resulting arrangement? Express your answer as a
reduced fraction.
7. William continually chooses numbers from 1 to 10, inclusive, with each having an equal probability. If
he picks a number bigger than 5, he stops. What is the probability that he will stop after choosing a 7?
8. Simon and Tyne each randomly choose a number between 1 and 10. What is the probability that the
non-negative difference between their numbers is at least 6? Express your answer as a percent.
9. Three coins, 2 fair and 1 unfair, are tossed. The probability that all three coins will end up on “heads” is
20%. What is the probability of 1 heads and 1 tails being tossed if the unfair coin were tossed twice by
itself?
10. The gelato shop “I Scream” offered up an irresistible “Random Bowl” promotion: out of 5 possible
flavors, the employees would choose a random number of flavors and would then choose the flavors
randomly. Given that there cannot be two scoops of the same flavor in a bowl, how many unique
“Random Bowls” are possible?
NRP Math Challenge Club
a. Alan says “I want N scoops of ice cream” but the employee pauses and says “We have a strict rule
saying all the scoops in your bowl must be different. I can’t give you that many scoops, I’m sorry.”
What is N?
11. A local lottery game gives players a shocking 60% chance of winning. Sean decides to play the game
for six straight days to see if he would win any money. What is the probability that he will win on at
least 3 out of the 6 days? Express your answer as a reduced fraction.
12.
In a best-of-7 series of games between two players, a person is declared a winner once that person
has won 4 games (no ties are allowed). In how many unique ways could the series play out?
BONUS:
1. Three perfectly logical men are told to stand in a straight line, one in front of the other. A hat is put
on each of their heads. Each of these hats was selected from a group of five hats: two identical
black hats and three identical white hats. None of the men can see the hat on his own head, and
they can only see the person’s hat in front of him. In how many distributions of the hats can the
person in front deduce his own hat color?
2. Five darts are thrown on a 2 m by 2 m dartboard. Prove that there exist two darts within √2
meters of each other.
3. Every set of N integers has a pair that differs by a multiple of 8. Determine the smallest value of N.
4. Andrew has a list of ten distinct 2-digit numbers. Michael and Justin want to pick numbers from
Andrew’s list and make their own sublists so that their lists have no values in common and the
numbers in their sublists have the same sum. Prove that they can always do this regardless of
Andrew’s initial list.
5. Funland is designed very strangely. It consists of N islands, connected by N-1 bridges so that you
can go from any one island in Funland to another using the bridges. What are some more
interesting facts about Funland?