Permutations and Combinations

A password has 3 letters (A–Z) followed by 2 digits (0–9). How many passwords can be made?

26×26×26×10×10 = 26³x10²

A restaurant offers 5 appetizers, 8 main courses, and 3 desserts. How many possible meals?

5×8×3

You roll a die 4 times. How many outcomes are possible?

6×6×6×6 = 6^4

You flip a coin 6 times. How many possible outcomes?

2×2×2×2×2×2 = 2^6

A locker combination uses 4 digits (0–9). Repeats are allowed.

10×10×10×10 = 10^4

A username has 2 letters (case-sensitive) followed by 4 digits.

• Case-sensitive letters: 26 uppercase + 26 lowercase = 52 choices per letter

• Digits: 10 choices (0–9)

• 52×52×10×10×10×10=52^2×10^4

A code has 2 digits and 2 uppercase letters

10×10×26×26

You choose a route with 3 turns, and each turn has 2 options. How many paths?

2×2×2 or 2³

A theater offers 5 movies, each at 3 times. How many total showtimes?

5×3

A school ID is made from 1 letter (A–Z), 3 digits (0–9).

26×10×10×10=26×1,000

How many ways to arrange the letters in the word “MATH”?

The number of arrangements is calculated as 4 factorial, or 4! = 24, since all letters are unique.

10 people compete in a race. How many ways to assign 1st, 2nd, and 3rd place?

10×9×8

How many ways can 4 books be arranged on a shelf from a set of 6 books?

P(6,4)=6×5×4×3

How many different 3-digit PINs can be made from digits 0–9 with no repeats?

10×9×8

A lock uses 5 unique letters. How many combinations?

P(26,5)=26×25×24×23×22

Choose and arrange 3 students from a group of 10 for 3 roles.

P(10,3)=10×9×8

4-letter password from A–F, no repeats:

P(6,4)=6×5×4×3

. Arrange 6 different math problems:

6!

Choose 5 cards from a 52-card deck

C(52,5)

5-member committees from 12 members

C(12,5)

Choose 6 lottery numbers from 1–49

C(49,6)