Q: A password consists of 3 letters followed by 2 digits. If there are 26 possible letters and 10 possible digits, how many different passwords are possible?

$$26^3$$ \times $$10^2 = 175$$,760$$

Question 1

A hacker is trying to guess a 5-character password using a keyboard with 92 possible characters for each position. What is the total number of possible passwords, and what is the probability of guessing the correct password on the first try?

Accepted Answer

$$92^5$$ possible passwords; probability = \frac{1}{$$92^5$$}$

Question 2

How many different ways can the letters in the word 'steam' be arranged?

Accepted Answer

$$5! = 120$$

Question 3

If the letters of 'steam' are arranged randomly, what is the probability that they will be in alphabetical order?

Accepted Answer

$$\frac{1}{120}$$

Question 4

Which of the following best describes the difference between permutations and combinations?

Accepted Answer

Permutations consider order, combinations do not.

Question 5

How many different trifecta bets are possible in a horse race with 20 horses, where you must select the first, second, and third place horses in the correct order?

Accepted Answer

$$\frac{20!}{(20-3)!} = 6840$$

Question 6

A survey has 10 questions, 2 of which are identical, and 3 others are also identical. How many different arrangements of the questions are possible?

Accepted Answer

$$\frac{10!}{2!3!} = 302,400$$

Question 7

In Florida’s Cash 5 lottery, you must select 5 numbers from 1 to 35. How many different lottery tickets are possible?

Accepted Answer

$$\frac{35!}{30!5!} = 324,632$$

Question 8

What is the probability of winning the Florida Cash 5 lottery with one ticket?

Accepted Answer

$$\frac{1}{324,632}$$

Question 9

Which formula would you use to determine the number of ways to arrange 8 books on a shelf if 3 of them are identical?

Accepted Answer

$$\frac{8!}{3!}$$

Question 10

A committee of 4 is to be formed from 10 people. How many different committees are possible?

Accepted Answer

$$\frac{10!}{6!4!} = 210$$

Question 11

Which of the following is the correct formula for the number of permutations of n items taken r at a time?

Accepted Answer

$$\frac{n!}{(n-r)!}$$

Question 12

If you have 7 different colored beads, how many ways can you arrange all of them in a row?

Accepted Answer

$$7! = 5040$$

Question 13

A password consists of 3 letters followed by 2 digits. If there are 26 possible letters and 10 possible digits, how many different passwords are possible?

Accepted Answer

$$26^3$$ 	imes $$10^2 = 175$$,760$$

Question 14

Which of the following statements about factorials is true?

Accepted Answer

$$0! = 1$$

Question 15

A pizza shop offers 5 toppings. How many different pizzas can be made with exactly 3 toppings?

Accepted Answer

$$\frac{5!}{2!3!} = 10$$

Counting Methods and Probability in Elementary Statistics

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