DSCI 501: Counting and Probability
1. What is the 10th term of ( x + y)50 ?
2. What is the coefficient of the xy7 term in (2x − 3y)8 ?
3. How many three digit numbers can you make with digits 1 to 9 without repetitions?
4. There is a lottery where a lottery ball machine with 20 balls is used to produce a
list of winning numbers from 5 balls. Which would make the lottery harder to win,
adding one more ball to the machine, or having the machine put out one more ball?
5. Austen, Sofia, and Isaac are ordering a pizza to share. Each person chooses a different single topping.
How many different ways could the three people order a pizza containing pepperoni, peppers, and onions? (Since each person is picking a different single topping,
one possible way would be: Austen picks peppers, Sofia picks onions, and Isaac
picks pepperoni.)(Note the question is about the order, not the physical layout of
the pizza.)
6. Austen, Sofia, and Isaac have realized that the order in which they choose pizza
toppings does not matter. Austen, Sofia, and Isaac order a new pizza. Each person
chooses a different topping out of five options. How many truly different pizzas
could the three friends order?
7. A 4-card hand is dealt from a deck of 52 cards. There are 52 × 51 × 50 × 49 =
6497400 ways that this hand can be dealt.
Once the cards are dealt, their order does not matter.
How many different 4-card combinations are possible from a deck of 52 cards?
8. Two committee co-chairs are chosen from a 10-person committee. How many cochair selections are possible?
9. Lukas has eight classes to choose from, and needs to take exactly four. How many
different selections of classes can Lukas choose to take?
10. A class has 30 students. Carrie wants to choose 5 students to do a project, and
Arthur wants to choose 25 students to do a project. Does Carrie or Arthur have
more choices?
11. Suppose you throw three standard dice: what is the probability of getting a sum of
4?
12. You are sitting in a group of 200 people at VIP seating at a concert. 20 of the people
in the VIP seating will be randomly chosen to go backstage. What is the probability
that you get to go backstage?
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13. Of 120 people at a lecture, 10 have red hair, 20 have blond hair, and the rest have
black or brown hair. If an attendee is selected at random, what is the probability
that their hair color is red or blond?
14. A die with 20 faces has each side labeled with a number from 1 through 10, with
some labels appearing multiple times. If the probability that the die lands on the
number 8 is 51 , how many sides are labeled 8?
15. When taking a standardized test, you are assigned to take the test in one of three
rooms. Two of the rooms hold the same number of people. One room holds three
times as many people as each of the other two rooms. What is the probability that
you will be assigned to the larger room?
16. There are two treasure chests, A and B. Treasure chest A contains 100 gold coins,
and treasure chest B contains 50 gold coins and 50 silver coins. You open a chest at
random, and draw a coin at random. If the coin is gold, what is the probability that
you opened chest A?
17. If three fair, six-sided dice are rolled, and the sum of the numbers rolled is even,
what is the probability that all three numbers rolled were even?
18. A disease test is advertised as being 99% accurate: if you have the disease, you will
test positive 99% of the time, and if you don’t have the disease, you will test negative
99% of the time. If 1% of all people have this disease and you test positive, what is
the probability that you actually have the disease?
19. In general, the probability that it rains on Saturday is 25%. Weekend rain has the following relationships: if it rains on Saturday, the probability that it rains on Sunday
is 50%. If it does not rain on Saturday, the probability that it rains on Sunday is 25%.
Given that it rained on Sunday, what is the probability that it rained on Saturday?
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