Show that if there are 30 students in a class, then at least two have last names that begin with the same letter

Q: How many outfits can be made from 4 pairs of slacks, 5 shirts, and 3 ties, if one selection from…

A: First decide on a pair of slacks.For that there are 4 different choices.Now for each of those…

Q: How many ways are there to arrange 12 identical apples and five different oranges in a row so that…

A: Here, arrange 12 identical apples and five different oranges in a row so that no two oranges will…

Q: Assume that there are 11 girls and 9 boys in the neighborhood club, and a team of 6 is to be…

A: As there are 11 girls and 9 boys now if a team of 6 is to be selected. 1) Number of different teams…

Q: For each of n = 0, 1, 2, 3, 4, how many 5-card hands from a 52-card deck contain exactly n queens?

A: To find- For each of n = 0, 1, 2, 3, 4, how many 5-card hands from a 52-card deck contain exactly n…

Q: If a phone company sends a random 6-digit number to customers to activate their phone, how many…

A: Consider a problem in which there are n vacant inline spots, each spot is to be filled by any of the…

Q: How many permutations of the letters OPQRSTU contain the strings QPO and PRS?

A: Solution

Q: assume that there 8 fire stations that need new chiefs and 12 qualified candidates In how many…

A: Given that, There are 8 fire stations Number of qualified candidates = 12

Q: Using the extended pigeonhole principle, show that if there are 30 students in a class, then at…

A:

Q: assume that there 8 fire stations that need new chiefs and 12 qualified candidates. In how many…

A: We have to find  number of ways..

Q: Suppose a set of 8 numbers are selected from the set: {1,2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14}…

A:

Q: Suppose there seven people at a party. Is it possible that everyone at the party knows exactly three…

A: It is not possible. Imagine constructing an acquaintance graph in which each of the 7 people are…

Q: How many ways to arrange 5 books on the shelf exist if two definite books are not placed together.

A:

Q: How many numbers must be selected from the set {2,4,6,8,10,12,14,16,18,20} to guarantee that at…

A: Use the principle of inclusion and exclusion to solve this problem, as in next step.

Q: Consider an alphabet that has 21 consonants and 5 vowels.  In how many ways can you make any…

A: This is from combinatorial.

Q: Find the number of permutations of the letters of these words:

A: Here permutations of word committee

Q: How many numbers must be selected from the set {1, 2, 3, 4, 5, 6} to guarantee that at least one…

A: To find  How many numbers must be selected from the set {1, 2, 3, 4, 5, 6} toguarantee that at least…

Q: Find the number of permutations using four different numbers (2,5,11,30) if they are taken 2 at a…

A: Given that, Four different numbers are (2,5,11,30) To find: Number of permutations using four…

Q: Show that if seven integers are selected from {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, there must be…

A:

Q: Assume that there are 9 pairs of shoes in a closet. How many single shoes must you choose at random…

A: Suppose the first shoe is chosen at random and after that 8 different shoes are chosen. Then the…

Q: Assume that you are assigning one of 12 available lockers to each of the 8 players on the team. In…

A:

Q: How many 3-digit numbers can be formed that are greater than 5 hundred and whose last digit is…

A: Given conditions: 3-digit number greater than 500 Last digit is a 3 or 9

Q: If repetitions are not permitted, how many 3-digit even numbers can be performed from the six digits…

A:

Q: How many unordered sets are there of two items chosen from four? sets

A: To Find: How many unordered sets are there of two items chosen from four? Concept: Crn=n!r!n-r!

Q: How many strings of eight uppercase English letters are there that start with X, if no letters are…

A: We have 26 possible letters in English alphabet: Now we have to form a string of eight uppercase…

Q: Show that if there are 8 people in a room, then at least two of them must have been born on the same…

A: There are 7 days in a week.  So we can have 7 people who can have different birthdays from Sunday to…

Q: Assume that there are 11 girls and 9 boys in the neighborhood club, and a team of 8 is to be…

A: It is the question of permutation and combination. And the whole solution is shown below.

Q: How many 4-letter strings of upper- and/or lowercase letters are there that contain at least two…

A: To determine the number of 4-letter strings of upper and /or  lowercase letters are there that…

Q: four largest 4-combinations o

A: Given: The set {1, 2, 3, 4, 5, 6, 7, 8} after {1, 2, 3, 5}. To find: The next four largest…

Q: How many three​-digit numbers may be formed using elements from the set​{1, 2, 3, 4, 5, 6, 7, 8, 9​}…

A: given:  set ​{1, 2, 3, 4, 5, 6, 7, 8, 9​}   if no element may be used more than once in a number and…

Q: Suppose that Ten service customers need service call and the maintenance worker can only service…

A: Based on the information, the dispatcher cares who serviced first, second and third. In this…

Q: suppose that ten service customers need service call and the maintenance worker can only service…

A: Consider the given information. Total Number of service customers = 10 Number of services done each…

Q: There are 15 persons in a party and if each 2 of them shakes hands with each other, how many ways…

A: Given data: 15 persons in a party Each 2 of them shakes hands with each other

Q: How many numbers must be selected from the set {1, 2, 3, 4, 5, 6, 7, 8} to guarantee that at least…

A:

Q: Suppose that two letters are to be selected from A, B, C, D, and E and arranged in order. How many…

A: Given: Two letters are to be selected from A, B, C, D, and E and arranged in order. There are in…

Q: a)How many four letter words including non-sense word can be formed from the word "BEAUTIFUL" if…

A: Given that  Word -   " BEAUTIFUL' Total alphabets in word = 9 Vowels = E , A , U , I , U = 5 vowels…

Q: In how many ways can 3 blue books and 4 red books be arranged on a shelf if a red book must be on…

A: The number of ways in which n different objects can be arranged, is given by n!, where,…

Q: How many unordered sets are there of four items chosen from six? sets

A:

Q: How many unordered sets are there of four items chosen from eight

A: According to the given information in this question We need to find the value of C(8,4)

Q: How many strings of eight uppercase English letters are there that start and end with X, if letters…

A: In English Alphabet we have 26 letters. We have to make a string of 8 uppercase English Letters.…

Q: suppose that five service customers need service call and the maintenance worker can only service…

A:

Q: By the Pigeonhole Principle Show that if there are 60 students in a class, then at least two have…

A:

Q: There are 100 people waiting in line for IDs.  Of these, 50 are from Indiana, 45 are freshman, and…

A: To count the number of persons satisfying the given conditions

Q: Suppose we want to choose 5 objects, without replacement, from 15 distinct objects.

A: Given data: The total numbers of objects are n=15. The numbers of objects choose is r=5. (a) Write…

Q: 5. Show that if there are 30 students in a class, then at least two have last name begin with the…

A:

Q: Consider 4 digit numbers with each digit from the set {1, 2, 3, 4, 5, 6, 7, 8, 9} .  How many such…

A:

Q: How many four-digit numbers can be formed from the numbers 1, 3, 4, 6, 8, and 9 if repetition of…

A:

Q: There are 12 teams in a league. In how many ways can the teams finish first, second, third, and…

A: A permutation is a mathematical tool that helps us to find the number of possible arrangements in a…

Q: How many ten-digit phone numbers can be formed from the digits 0,1,2,3,...,9 if the first digit may…

A: To find: No of ten-digit phone numbers can be formed from the digits 0,1,2,3,...,9 if the first…

Q: 2) How many numbers are there between 6000 and 7000 which have exactly one of their digits as 7?

A: Since, we have to find numbers between 6000 and 7000 so the thousand's place must be occupied by 6.…

Q: How many different ID cards can be made if there are 6 digits on a card and no digur can be used…

A: Given: Number of digits = 6 No digits is used more than once.

Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.