Practice Problems on Counting Principles and Probability
1. Out of the digits 0,1,…9, how many 4 digit numbers can you form so that …
a. The numbers are odd.
b. The numbers have distinct digits.
c. The numbers are divisible by 5.
d. The numbers are divisible by 10.
2. You are coaching an amateur softball teams consisting of 15 men and 8 women. It
is a slow pitch, recreational league, so that we have 10 spots on a batting order
(you got an extra “rover”, a player in between infielders and outfielders). How
many ways to construct a batting order if….
a. Women don’t bat.
b. The leadoff (1st batter) must be a woman.
c. The cleanup (4th batter) must be a man.
d. 1st thru 4th are men and 8th thru 10th are women.
e. Men and women bat alternatively (i.e. a man must bat after a woman, and vice
versa)
3. You come to a pizza restaurant and order a pizza of 6 slices. You can choose any
topping on each slice. The available toppings are Pepperoni, Sausage, Olive,
Pepper, Mushroom, Chicken, Bacon, Pineapple, Onion, Tomato, Jalapeno. We
want every slice to have different toppings.
a. How many choices of pizzas can you order?
b. How many choices can you order if you have exactly 1 slice with meat?
c. How many choices can you order if you have exactly 2 slices with meat?
d. How many choices can you order if there is no meat at all on the pizza?
e. Pick your toppings completely randomly. What is the probability of having a
pizza with at least 1 slice of meat topping?
4. In poker game, a “Full house” is when you have 5 cards, 3 of them have the same
numbers, and the other 2 cards have the same numbers. (e.g. 44422, 555KK,
JJJ22 … etc).
a. When picking up five cards randomly, how many possible combinations can
you have?
b. How many possible ways to obtain a full house?
c. What is the possibility to obtain a full house when five cards are drawn
randomly?
“Four of a kind”, out of five cards, is the combination of having 4 of your 5 cards
having the same number (e.g. 33335, AAAA2 … etc)
d. When drawing five cards, how many ways to obtain “four of a kind”?
e. What is the possibility of obtaining “four of a kind” when drawing 5 cards
randomly?
f. How do c. and e. compare? Does it fit your expectation? (Which combination
beat the other one?)