License plate numbers in a certain state consists of seven characters. The first character is a digit (0 through 9). The next four characters are capital letters (A through Z) and the last two characters are digits. Therefore, a license plate number in this state can be any string of the form: Digit-Letter-Letter-Letter-Letter-Digit-Digit (a) How many different license plate numbers are possible? (b) How many license plate numbers are possible if no digit appears more than once? (c) How many license plate numbers are possible if no digit or letter appears more than once?

Q: In the local boating pond there are 11 plain wooden boats, each with a different number. The owner…

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Q: The records of 1492 high school students were examined, and the following information was obtained:…

A: The records of 1492 high school graduates were examined, which means that the universal set, Now,…

Q: Your friend gives you a list of three five-digit numbers, but she only reveals one digit in each: 3…

A: as we have given three five digit number 3 ? ? ? ?? 8 ? ? ? ? ? 2 ? ? claim- check whether we can…

Q: In a certain state, each automobile license plate number consists of two letters followed by a…

A: There are 24 choices for each letter. (this explanation was already shown in the image you shared)…

Q: A jeweler selects 3 gems for a ring. The jeweler has: diamonds, sapphires, and emeralds. Let’s use…

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Q: Mr. H has a collection of 500 vessels. In how many ways can he name his favorite and second favorite…

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Q: The Liberty County Poetry Society awarded 3 different prizes 1 each for 1st, 2nd, and 3rd place…

A: The first prize can be given in 56 ways Second prize in 55 ways Third prize in 54 ways

Q: In choosing what music to play at a charity fund raising event Cory needs to have an equal number of…

A: If an event can be done in n different ways and an other event can be done in m different ways, then…

Q: Arealtor uses a lock box to store the keys to a house that is for sale. The access code for the lock…

A: The access code contains 4 digits          Digits : 0,1,2,3,4,5,6,7,8,9 The first digit can…

Q: access code for the entrance of a faculty office consists of five digits. Each digit can be 0…

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Q: You are asked to create a six character password and each character can be any of the 26 letters of…

A: Given: Number of alphabets=26 Number of numerical 2 through 9=8 Total number of characters, n=34 The…

Q: Bob owns 13 different books. He wants to pick out and arrange 5 of these books left-to-right on a…

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Q: Three-letter call signs In 1912 the U.S. government began issuing licenses to radio stations. Each…

A: Given information: All the call signs beginning with W, K, and N are reserved by US.

Q: How many different​ 4-letter acronyms can be made if the first letter must be K, U, or Y and no…

A: How many different 4-letter acronyms can be made if the first letter must be K, U, or Yand no letter…

Q: a) How many plates are possible when letters and numbers can repeat? Set up the expression that can…

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Q: how many ways are there to arrange the letters in the word BAGGAGE.

A: The objective of this question is to find out the number of ways the letters in the word 'BAGGAGE'…

Q: Solve the following system of differential equations with the given initial conditions, and then, to…

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Q: is granted, how many different ways are there to schedule the appearances? ways (Type a whole…

A: It is given as the 6 performers who will present their comedy acts this weekend at a comedy club.

Q: Bob owns 7 different books. He wants to arrange all of these books left-to-right on a single…

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Q: 200 people yielded this information: 94 people own a blu-ray player 127 people owned a microwave…

A: Given: 200 people yielded this information: 94 people own a blu-ray player 127 people owned a…

Permutations and Combinations

If there are 5 dishes, they can be relished in any order at a time. In permutation, it should be in a particular order. In combination, the order does not matter. Take 3 letters a, b, and c. The possible ways of pairing any two letters are ab, bc, ac, ba, cb and ca. It is in a particular order. So, this can be called the permutation of a, b, and c. But if the order does not matter then ab is the same as ba. Similarly, bc is the same as cb and ac is the same as ca. Here the list has ab, bc, and ac alone. This can be called the combination of a, b, and c.

Counting Theory

The fundamental counting principle is a rule that is used to count the total number of possible outcomes in a given situation.