Using the extended pigeonhole principle, show that if there are 30 students in a class, then at least 2 of them have names starting with the same alphabet.

Q: How many passwords with 5 letters without repetition can be made from letters a,b,c,d,e,f,g,h,I,j

A: Solve using permutations

Q: Registrants at a large convention are offered 6 sightseeing tours on each of 3 days. In how many…

A: Given: Number of sightseeing offered is, n=6. Number of days is, d=3. The objective is to find the…

Q: How many distinct ten-digit passwords are possible if you must use at least one 26 lower case letter…

A: Given: you must use at least one 26 lower case letter from {a, b, . . . , z}, at least one 26 upper…

Q: Eight horses are entered in a race. In how many ways can the horses finish considering that at least…

A: Permutation: The number of different arrangements of ‘r’ elements from a set of‘n’ elements is…

Q: How many strings of length two can be formed by using the letters A, B, C, D E and F without…

A: Given letters, A, B, C, D, E, F We have to find number of strings of length two

Q: How many 5 letter "words" can be formed if exactly 2 of the letters are vowels?  Condition: order is…

A: Let there are three persons A, B and C sitting beside each other. The different ways in which this…

Q: Suppose you are at a party with 19 of your closest friends (so including you, there are 20 people…

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Q: how many ways are there to pick the set of people who are interviewed

A: We have given that  Total number of applicants = 50 After reduced the number of candidates than the…

Q: How many 3-digit number greater than 200 can be formed (a) repetition not allowed (b) repetition…

A:

Q: How many five-letter passwords can be formed from the letters {A, B, J, K, X, Z} under the following…

A: From the provided information, The letters are {A, B, J, K, X, Z} Total letters = 6

Q: how many ways can you invite at least 7 of your 10 friends?

A: Given that  Total number friends (n) = 10 Number of friends to invite = At least 7    We know that…

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

A: We are given that there are 30 students in the class. We will assume that every student has first…

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

A: Solution

Q: In a certain state license plates are formed by choosing 2 letters followed by 4 digits without…

A: Solution: Let L1 and L2 be the first two letters of a license plate.  For L1 we have 26 options…

Q: 5. How many strings of 3 characters are there, such that every character is a lower case letter, the…

A: If there are a ways of doing something and b ways of doing another thing, then there are a · b 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}…

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Q: how many combinations are possible? assume the items are distinct. 16 items chosen 16 at a time

A: Only one combination is possible.

Q: How many combinations are possible? Assume the items are distinct. 5 items chosen 3 at a time…

A:

Q: same

A:

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: The gift wrap desk at a large dept. store offers 5 box sizes, 10 wrapping papers, 7 colors of ribbon…

A: To calculate the total number of options for the customer at the gift wrapping counter

Q: How many ways are there to invite one of four different friends over for dinner on five successive…

A: Let x, y ,z, u be the friends and a, b, c, d denotes the number of times a friend is invited…

Q: How many strings of seven lower case letters of the English alphabet contain exactly four letters w…

A: We have to use permutations and combinations here.

Q: In an experiment on social​ interaction, 6people will sit in 6 seats in a row. In how many ways can…

A: Here we Wang to know how many ways six people can sit in six seats in a row.

Q: How many ways are there for 6 people sitting around a circular table to pick from 12 entrees so that…

A:

Q: 8 athletes are entered in a race. How many possible ways can they finish (assuming no ties are…

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Q: on a basketball team be filled with 10 In how many ways can 5 starting players (no positions)…

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Q: A license plate consists of   five symbols on each​ plate, where the first two symbols are letters…

A: Fundamental principle of counting : If an event occur in m ways an corresponding to it another event…

Q: How many permutation of the letter ABCDEFGH contain the string ABC?

A:

Q: Find the number of ways of getting at least a pair when randomly getting nine cards from a standard…

A:

Q: A child wants to pick 5 balloons from a set of 6 blue and 5 red balloons. Number of ways for picking…

A: Number of blue balloons = 6 Number of red balloons = 5

Q: Registrants at a large convention are offered 6 sightseeing tours on each of 3 days. In how many…

A: Number of sightseeing tours for one day are 6. Number of days for the tour is 3.

Q: Find the total possible number of 5 letter words such that they are alphabetically first in the…

A:

Q: Find the number of distinguishable permutations of the digits of the number 228,434.

A: Write the number in frequency format: Numbers Frequency fi 2 2 8 1 4 2 3 1 Total…

Q: A PIN-number for a credit card consists of a sequence of 4 digits chosen from the set…

A: Given the set {0,1,2,3,4,5,6,7,8,9} We find the 4 digits different PIN number

Q: A class has 11 students. What is the probality that exactly 4 of the students were born on a weekday…

A: We have to find given probability.

Q: How many words can be formed by permutation of the letters of the word DEPRESSION such that each…

A: Given word DEPRESSION We have to calculate No. Of words such that obtained word does not contain 2…

Q: How many different ways can a manager and supervisor select from an IT company branch in Manila if…

A: Query- How many different ways can a manager and supervisor select from an IT company branch in…

Q: Using the pigeonhole principle How many numbers must be selected from the set {1, 3, 5, 7, 9, 11,…

A:

Q: There are 29 students in precalculus; in how many ways can students fill the Top five?

A: Permutation: A permutation is an arrangement of objects in a definite order.   Let n is the total…

Q: How many different words can be formed with the letters of the word ‘SUPER’ such that the vowels…

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Q: students be invited, then compute the number of ways undergraduate students are invited.

A: Here use combination

Q: set of two - digit numbers obtained from the digits 1

A:

Q: How many words can be formed by permutation of the letters of the word DEPRESSION such that each…

A:

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

A:

Q: How many ways can a local chapter of the American Chemical Society Schedules 3 speakers for 3…

A: As Dr. S. N. De said "A permutation of a set is defined as an arrangement of its members into a…

Q: How many strings of 5 upper case letters of the English alphabet start or end with A? Letters could…

A: detailed explanation is mentioned below

Q: When purchasing a ticket for the PowerBall lottery game a player chooses 5 distinct numbers between…

A:

Q: How many combinations are possible to select a student president, vice president and treasurer among…

A: