Counting + Casework
How many three-digit whole numbers are there?
There are 900: 100 - 999.
120 and 20.
6!/(6-3)! = 6*5*4 = 120
6!/(3!*(6-3)!) = 20
What is the probability of selecting 2 Aces from a shuffled deck without replacement?
1/221
4/52 * 3/51 = 1/13 * 1/17 = 1/221
Hanning has won a prize at the fair and gets to choose 4 different prizes out of a set of 7. How many combinations of 4 prizes can she choose?
7C4 = 35.
Grayson has two pennies and three nickels in her piggy bank. If she randomly picks out a coin, what is the probability that it will be a penny?
2/5
How many whole numbers less than 100 are multiples of 3 but NOT multiples of 5?
27 (33 multiples of 3 - 6 multiples of 15)
Annie, Ben, Clara, Donald, and Eric are sitting in a row at the theater. How many different seating arrangements are there for the five friends?
5! = 120.
6 students of different ages are randomly seated in the first row of a classroom. What is the probability that from left to right they are seated oldest to youngest?
1/6! = 1/720
16 points are placed on the circumference of a circle. How many lines will it take to connect every point to every other point?
16*15/2 = 120 lines
16C2 = 120.
Seven runners are running a marathon. How many ways can three of the runners place 1st, 2nd, and 3rd?
7 * 6 * 5 = 210 ways
A class of 40 students is comparing pets. If 22 students have a dog and 30 students have a cat, what is the fewest number of students that can have both a dog and cat?
At least 12 students must have both a dog and a cat.
22 + 30 - 40 = 12 students
Find the expression for the number of arrangements of the letters in the word BOOKKEEPER.
10!/(2! * 2! * 3!) = 151200
What is the probability that the top card in a shuffled deck is a red Ace and the second card is a spade?
1/102
2/52 * 13/51 = 1/102
Matt wants to sell plushies at the school store. He has only 8 plushies to sell. How many different combinations of 3 plushies can Hanning choose to buy?
8C3 = 56 combinations
How many positive three-digit integers are multiples of 6?
First team to answer correctly gets the points!
150
Several pairs of twins arrive at a dinner party. Each person at the party shakes the hand of every other person, not including their twin. If there were a total of 112 handshakes, how many sets of twins attended the party?
8 pairs of twins.
If n is the number of people who attended, the total handshakes are n(n-2)/2 = 112. So, n = 16.
How many arrangements of the letters in the word BEGGING have an N at the beginning?
120 arrangements
There are 6! ways to arrange the letters BEGGIG, but we need to divide by 3! to account for the repeated G.
In six rolls of a standard die, what is the probability that the same number will be rolled exactly 5 times? Give the expression and do not evaluate.
5 * 6 * 6 / 6^6.
Paul flips a fair coin 6 times. In how many ways can he flip at least 2 heads?
2^6 - 1 - 8 = 55 ways.
There are seven parking spaces in a row, and four must be reserved for VIP's. How many ways can the spaces be reserved if at least two of the reserved spaces must be adjacent?
(Use complementary counting!!)
7C4 - 1 = 34.
A fair coin is slipped 10 times. Write the expression for the number of ways to flip MORE heads than tails.
2^10 - 10C0 - 10C1 - 10C2 - 10C3 - 10C4 - 10C5.
In how many arrangements of the letters in the word EXAMPLE are the letters A and M next to each other?
6! * 2 / 2 = 720.
There are 3 vegetarians in a class of 20 students. If 2 students are chosen at random, what is the probability that exactly one of the three is a vegetarian? Give the expression and do not evaluate.
3C1 * 17C2 / 20C3
Anna walks 4 blocks on a city grid of sidewalks from home to work every day: 2 blocks north and 2 blocks west. She never uses the exact same path on her return trip home. If Anna always stays on the city sidewalks and walks 4 blocks every day, how many ways can she walk to work and back?
30 ways
There are 4C2 = 6 ways (NNWW) for Anna to walk to work. There are 5 ways for her to walk back. She can walk 6*5 = 30 ways to work and back.
Zip codes in the US are five digits long, followed by a four-digit code (ex. 27134-1112). In North Carolina, every zip code begins with either 27 or 28 (ex. 28122-1023). How many 9-digit zip codes are possible in North Carolina in which each digit is only used once? Write the expression and do not evaluate.
First team to answer correctly gets the points!
8! * 2