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How many different 4-letter passwords can be created using the letters A, B, C, D with repetition allowed?

There are $4^4 = 256$ different 4-letter passwords.

How many different arrangements are there of the word 'MATHEMATICS'?

$rac{11!}{2!2!2!} = 4,989,600$ different arrangements.

How many 10-digit telephone numbers can be formed if the first digit cannot be 0 or 1?

$8 imes 10^9$ different telephone numbers can be formed.

For a 3-question multiple-choice test where each question has 4 options, how many different ways can a student answer the test?

$4^3 = 64$ different ways to answer the test.

If a committee of 4 must be selected from 12 candidates, how many ways can this be done?

$C(12, 4) = rac{12!}{4!(12 - 4)!} = 495$ ways.

How many 6-digit numbers can be formed with the digits 1, 2, 3, 4, 5, 6 if no digit can be used more than once?

$P(6, 6) = 6! = 720$ different 6-digit numbers.

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

There are 5! = 120 ways to arrange the letters in the word 'COUNT'.

How many ways can you choose 2 appetizers, 3 main courses, and 1 dessert from a menu with 6 appetizers, 8 main courses, and 4 desserts?

$C(6, 2) imes C(8, 3) imes C(4, 1) = 15 imes 56 imes 4 = 3,360$ ways.

How many ways can you distribute 5 distinct toys to 4 children if each child can receive no toys or as many as all the toys?

$4^5 = 1,024$ ways to distribute the toys.

A PIN consists of 4 digits followed by 2 letters. How many different PINs can be formed?

$10^4 imes 26^2 = 6,760,000$ different PINs can be formed.

How many three-letter words, with or without meaning, can be formed from the English alphabet when repetition of letters is allowed?

$26^3 = 17,576$ different words can be formed.

A lock combination uses 3 numbers, each from 0 to 9. How many possible lock combinations are there if numbers can be repeated?

$10^3 = 1,000$ possible lock combinations.

If there are 15 teams in a tournament, how many games must be played so that each team plays every other team exactly once?

$C(15, 2) = rac{15!}{2!(15 - 2)!} = 105$ games must be played.

In how many different ways can a president, vice-president, and treasurer be elected from a club of 10 people?

$P(10, 3) = rac{10!}{(10 - 3)!} = 720$ different ways.

You have 3 pants and 5 shirts. How many different outfits consisting of 1 pant and 1 shirt can you wear?

You can have 15 different outfits.

In a race with 8 runners where there are no ties, in how many different ways can first, second, and third place be awarded?

$P(8, 3) = rac{8!}{(8 - 3)!} = 336$ different ways.

How many unique ways can the letters of the word 'LEVEL' be arranged?

$rac{5!}{2!2!} = 30$ unique ways.

How many unique ways can 5 paintings be lined up on a wall?

There are $5! = 120$ unique ways to arrange the paintings.

A restaurant offers 4 starters, 5 main courses, and 3 desserts. How many different meals consisting of one starter, one main course, and one dessert are there?

There are $4 imes 5 imes 3 = 60$ different meal combinations.

How many 5-card hands can be dealt from a standard 52-card deck?

There are $C(52, 5) = rac{52!}{5! (52 - 5)!} = 2,598,960$ possible 5-card hands.

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