License Plates
An automobile license plate is made up of 2 letters followed by 3 digits. The first and third digits cannot be a 0. The third digit also cannot be a 9. With these restrictions, how many different license plates were possible?
486,720
How many 11 digit numbers can we make without repetitions?
39,916,800
In how many ways can you select a committee of 3 students out of 10 students?
120
A biology teacher has 5 different pets they keep in their classroom. For an upcoming holiday break, the teacher will send the pets home with students. Suppose that 16 of the teacher's 75 students volunteer to take a pet home, and the teacher will randomly select 5 of those volunteers to each take 1 pet home.
The permutation formula can be used to find the number of unique ways the teacher can distribute pets to the volunteers.
If there are 9 female volunteers and 7 male volunteers, what is the probability that 2 females and 3 males would be chosen?
0.0288
Amari has 8 paint colors available, and he wants to mix 3 of them to create a new color.
How many different sets of 3 colors can Amari choose?
56
In a certain country, the car number plate is formed by 2 digits from the digits 1, 2, 3, 4, 5, 6, 7, 8 and 9 followed by 5 letters from the alphabet. How many number plates can be formed if neither the digits nor the letters are repeated?
710,424,000
A youth baseball team has 13 players. How many different ways are there for the coach to select and arrange 9 of these players to make up the team’s batting order?
259,459,200
A basketball team has 10 players. Before each game, the coach picks 2 of those players to carry the team's water jug.
How many different groups of 2 players can the coach pick?
45
Lin and Kai are friends that work together on a team of 12 total people. Their manager is going to randomly select 2 people from the team of 12 to attend a conference.
What is the probability that Lin and Kai are the 2 people chosen?
1/66 = 0.0152
A club is going to send 4 of its 10 members to represent the club at a conference.
How many different groups of 4 members can they send?
210
The standard license plate for California passenger cars has one digit, followed by four letters, and then two more digits. The first digit cannot be a 0 or 3. The first and third letters cannot be O or Q. How many possible license plates are there?
11,980,800
How many 6 letter words can we make using the letters in the word LIBERTY without repetitions?
720
Manny's burrito stand lets customers choose 4 ingredients from the list below to add their burritos:
onions, cilantro, jalapeño, beans, cheese, lettuce, salsa, hot sauce
How many groups of 4 different ingredients can go on a burrito?
70
Michael is taking a quiz in his music history class. The teacher writes the names of 6 songs on the board, and then plays 4 songs out of the 6, one after the other in a random sequence. Michael then needs to identify each of the songs in the sequence they were played.
Suppose that Michael hasn't studied and is randomly guessing the songs.
What is the probability that Michael correctly identifies all 4 songs in the correct order?
1/360 = 0.002778
In a certain country, the car number plate is formed by 4 digits from the digits 1, 2, 3, 4, 5, 6, 7, 8 and 9 followed by 3 letters from the alphabet. How many number plates can be formed if neither the digits nor the letters are repeated?
47,174,400
A standard-issue 2020 New Jersey license plate had one letter, then two digits, followed by three letters. These plates were not allowed to have the letters I, O, D, T, X, or Q in any position. The first letter also cannot be a P. The first digit cannot be a 2. With these restrictions, how many different license plates were possible in 2020?
13,680,000
You need to put your reindeer, Jebediah, Lancer, Ezekiel, Gloopin, and Bloopin, in a single-file line to pull your sleigh.
120
A committee including 3 boys and 4 girls is to be formed from a group of 10 boys and 12 girls. How many different committee can be formed from the group?
59,400
Recently, a company had to fire 12 employees. The company claimed that it selected the employees who were fired at random. Of the 12 employees fired, 6 were women (50%). However, only 10 of the company’s 30 employees were women (33%).
Find the probability that a random selection of 12 employees from the company would result in the selection of 5 women and 7 men.
0.2259
Five runners need to be arranged in 5 lanes for a race so that there is a runner in each lane.
How many unique ways are there to arrange the runners in the 5 lanes?
120
A standard-issue license plate had two letters, then one digit, followed by four letters. These plates were not allowed to have letters D or T in the second position, or the letters I, O, or Q in any position. The digit cannot be a 0. With these restrictions, how many different license plates were possible?
1.216*10^9 or 1,216,000,000
Seven runners are competing in a race where 3 of them will earn medals for finishing first, second, and third.
How many unique ways are there to arrange 3 of the 7 runners in first, second, and third place?
210
Recently, a company had to fire 12 employees. The company claimed that it selected the employees who were fired at random. Of the 12 employees fired, 6 were women (50%). However, only 10 of the company’s 30 employees were women (33%).
Find the number of ways in which a random selection of 12 employees from the company would result in 6 women and 6 men being selected.
8,139,600
A standard deck of 52 playing cards – 4 cards each of 13 different denominations (ace, two, three,…,ten, jack, queen, king). You and a friend decide to play a game in which you each start with 5 cards. Suppose you shuffle the deck well and deal out 5 cards to you and your friend. You are surprised when you look at your hand and see three aces (along with two non-aces).
Find the probability of dealing yourself 3 aces and 2 non-aces in a 5-card hand from a well-shuffled deck.
0.00174
The Transportation Security Administration (TSA) is responsible for airport safety. On some flights, TSA officers randomly select passengers for an extra security check prior to boarding. One such flight had 76 passengers – 12 in first class and 64 in economy class. Some passengers were suspicious when all 10 passengers chosen for screening were seated in economy class.
Use combinations to find the probability that all 10 passengers chosen at random for security screening are in economy class.
0.159