Q: A necklace is made with 5 red beads, 4 blue beads, and 3 green beads. How many distinguishable ways can the beads be arranged?

$$12!/(5! \times 4! \times 3!)$$

Q: A restaurant offers 12 appetizers, 15 main courses, and 10 desserts. If you must choose 2 appetizers, 3 main courses, and 1 dessert, how many different meal combinations are possible?

$$(12C2) \times (15C3) \times (10C1)$$

Q: A bag contains 7 red marbles, 5 blue marbles, and 3 green marbles. How many distinguishable ways can all the marbles be arranged in a row?

$$15!/(7! \times 5! \times 3!)$$

Q: A bag contains 4 red, 5 blue, and 6 green candies. If you want to arrange all candies in a row, how many distinguishable arrangements are possible?

$$15!/(4! \times 5! \times 6!)$$

Question 1

A school has 12 different books. How many ways can all 12 books be arranged on a shelf?

Accepted Answer

$$12!$$

Question 2

A committee of 5 students is to be chosen from a group of 20. If the order in which the students are chosen does not matter, how many different committees are possible?

Accepted Answer

$$20C5$$

Question 3

A password consists of 4 unique digits. What is the probability of correctly guessing the password on the first try?

Accepted Answer

$$\frac{1}{4!}$$

Question 4

How many distinguishable ways can you arrange the letters in the word 'BALLOON'?

Accepted Answer

$$7!/(2! 	imes 2!)$$

Question 5

A pizza shop offers 8 toppings. If you want a pizza with 3 different toppings and the order does not matter, how many possible pizzas can you make?

Accepted Answer

$$8C3$$

Question 6

In a race with 10 runners, how many ways can the top 3 finishers be arranged?

Accepted Answer

$$10P3$$

Question 7

A necklace is made with 5 red beads, 4 blue beads, and 3 green beads. How many distinguishable ways can the beads be arranged?

Accepted Answer

$$12!/(5! 	imes 4! 	imes 3!)$$

Question 8

A restaurant offers 12 appetizers, 15 main courses, and 10 desserts. If you must choose 2 appetizers, 3 main courses, and 1 dessert, how many different meal combinations are possible?

Accepted Answer

$$(12C2) 	imes (15C3) 	imes (10C1)$$

Question 9

A lottery requires you to select 6 numbers from a set of 49. What is the probability of winning if only one combination is correct?

Accepted Answer

$$\frac{1}{49C6}$$

Question 10

A bag contains 7 red marbles, 5 blue marbles, and 3 green marbles. How many distinguishable ways can all the marbles be arranged in a row?

Accepted Answer

$$15!/(7! 	imes 5! 	imes 3!)$$

Question 11

A team of 4 students is to be selected from a class of 18. If the order of selection is important, how many possible teams can be formed?

Accepted Answer

$$18P4$$

Question 12

A medical research team needs to select 3 patients from a group of 25 for a new treatment. If the order does not matter, how many ways can the patients be chosen?

Accepted Answer

$$25C3$$

Question 13

A school awards medals to the top 4 students in a math competition. If there are 14 students competing, how many ways can the medals be awarded?

Accepted Answer

$$14P4$$

Question 14

A group of 6 friends wants to sit in a row at a movie theater. If two of the friends insist on sitting together, how many possible seating arrangements are there?

Accepted Answer

$$5! 	imes 2!$$

Question 15

A scientist needs to select 2 contaminated and 3 non-contaminated samples from a set of 7 contaminated and 15 non-contaminated samples. How many ways can this selection be made?

Accepted Answer

$$(7C2) 	imes (15C3)$$

Question 16

A code consists of 5 unique letters. What is the total number of possible codes?

Accepted Answer

$$5!$$

Question 17

A contest awards prizes to the top 3 out of 12 participants. If the order of prizes matters, how many ways can the prizes be awarded?

Accepted Answer

$$12P3$$

Question 18

A bag contains 4 red, 5 blue, and 6 green candies. If you want to arrange all candies in a row, how many distinguishable arrangements are possible?

Accepted Answer

$$15!/(4! 	imes 5! 	imes 6!)$$

Question 19

A university is forming a committee of 3 professors from a department of 10. What is the probability that the committee consists of Professors A, B, and C?

Accepted Answer

$$\frac{1}{10C3}$$

Permutations, Combinations, and Factorials: Advanced Counting in Probability 3.4

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