Remember that we can write n as a sum of n ones. How many plus signs do we use? Question: In how many ways may we write n as a sum of a list of k positive numbers? Such a list is called a composition of n intok parts. For example, there are three different ways can we write n=5 as a sum of a list of k=3 positive numbers: • 1+1+3 = 5 1+3+1 = 5 • 3+1+1 = 5 Notice that we want to count all three sums, even though they all use two "1's" and one "3". HINT: How many ways can you insert two "+" signs into the list 1 1 111?

Q: Farmer Brown owns five horses and six mules. He takes them into the barn where there are five stalls…

A: Given that, Farmer Brown owns five horses and six mules. He takes them into the barn where there are…

Q: How many integers from 1 through 1000 are multiples of 7 or multiples of 4?

A: Let A be the set of integers that are multiples of 7 and B be the set of integers that are multiples…

Q: uppose we want to choose 2 letters, without replacement, from the 5 letters A, B, C, D, and E. (a)…

A: It is given that 5 letters A, B, C, D and E.

Q: how many way can six books be arranged on a shelf

A: We need to find the number of ways for arrangement of six books on a shelf.

Q: How many numbers of four different digits each greater than 5000 can be formed from the digits 1, 2,…

A: Answer - How many numbers of four different digits each greater than 5000 can be formed from the…

Q: How many different ways can the letters of "committee" be arranged?

A:

Q: In how many ways can 9 gifts be divided between 4 children if the eldest child is to receive 3 gifts…

A:

Q: How many ways can 4 books be arranged on a shelf if the books are selected from a collection of 100…

A: Here we will use the concept of permutation Permutation = nPr = n!/(n-r)!

Q: Please use permutations in your solutions, these are all related to permutations, so I hope…

A: Given:  group 10 people using permutation how many different ways to choose the officers

Q: An auditorium has 28 rows of seats. There are n=24 seats in the first row, 25 seats in the second…

A: Given that an auditorium has 28 rows.  Number seats in first row= 24  Number seats in second row= 25…

Q: How many ways can all the letters in the word EDUCATION can be arranged if all the vowels are right…

A:

Q: a. The large books must always be together. b. The first book and last book must always be medium…

A:

Q: How many different ways can the letters of "jogging" be arranged? The number of different ways that…

A: The word “jogging” contains a total of 7 letters in which “g” letter is repeated 3 times.

Q: How many ways can you write a 5 digit number such that some of the digits repeat? (Note: 010 is a 2…

A: Given query is to find the ways to write 5 digit number.

Q: in hpw many ways can we arrange 6 cats followed by 4 dogs in a line if there are 9 cats and 6 dogs…

A: arrange 6 cats followed by 4 dogs in a line if there are 9 cats and 6 dogs to choose from.

Q: How many unique ways can the letters for M-A-X-I-M-U-M be arranged?

A: To find the number of ways that the letters can be arranged.

Q: Explain why an odd counting number can always be written in the form 2N+1 for some whole number N.

A: The objective of this question is to understand why any odd counting number can be expressed in the…

Q: Bob owns 9 different books. He wants to arrange all of these books left-to-right on a single…

A: From the provided information, Bob owns 9 different books. And he wants to arrange these books left…

Q: Bob owns 15 different books. He wants to pick out and arrange 4 of these books left-to-right on a…

A:

Q: How many terms are in the sum: 10 + 21 + 32 + 43 + .. + 549? Note: a = 10.

A:

Q: how many ways can a teacher sit 12 students in two rows of 6 chairs if Bobby must sit in the front…

A: Given: Total number of students = 12 Number of seats in each row = 6 Bobby is in first row and Billy…

Q: How many 4 digit numbers can we make using the digits 2, 5, 7, and 9 without repetitions? O…

A:

Q: 2. Find the sum of the following terms: 17b?-33b?+105b?+62b?-109b?-5b?.

A: Given: 17b2-33b2+105b2+62b2-109b2-5b2

Q: For a segment of a radio show, a disc jockey can play 8 records. If there are 13 records to select…

A:

Q: How many ways can 7 people fall in line if 2 persons doesn’t want to be seated next to each other?

A:

Q: Natasha and Rodrigo are in a class of 30 students that selects 4 leaders. How many ways are there to…

A:

Q: For a segment of a radio show, a disc jockey can play 9 records. If there are 14 records to select…

A: For a segment of radio show, a disc jockey can play 9 records.If there are 14 records to select…

Q: In how many ways 7 boys and 6 girls can be arranged on a bench if Anne and Jim sit together?

A:

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

A:

Q: How many ways can you park cars in 5 parking spots if there are 15 cars to choose from? You may…

A: 15 cars can be parked in 5 parking spots in C515ways. So, the number of possibilities = C515= 3003…

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.