Show that if seven integers are selected from {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, there must be a pair among the selected integers whose sum equals 13.

Q: Suppose that in a class, there are n many boys and n many girls. Among these 2n many people, we will…

A: Let A be a set of n elements. To select r objects from a collection of n elements in the set A…

Q: 1. How many 5 digit positive integers are divisible by 5 and have at least one 6 as their digit?

A: A number is divisible by 5 if the last digit is either 5 or 0. We have to find the number of 5 digit…

Q: Compute the sum of the first 60 positive integers that are exactly divisible by 4.

A: The numbers divisible by 4 is:

Q: If ten integers are chosen from between 0 and 21 exclusive, must at least one of them be even?…

A: Pigeon hole principle states that if n objects are put in to m containers with n>m then at least…

Q: How many six-digit positive integers contain exactly one one, two twos and three treor? How many…

A: First note that here six digit integer can start only with 1 or 2. If it starts with 1 the remaining…

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 strings of 1our decimal digits have all four digits that are different? How many strings of…

A:

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: Let [A1, A2], [B1, B2], [C1, C2] be 3 married couples who attended a concert.  In how many ways can…

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

Q: How many four digit positive integers are there that contain the digit 3 and are divisible by 5?

A: A number is divisible by 5, if the unit digit place contains the number 0 or 5. Let m choices to be…

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}…

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Q: Determine the number of permutations of the set {1 ,2 ,..., 14} in which exactly 7 integers are in…

A: Solution-:

Q: If seven integers are chosen from between 1 and 12 inclusive, must at least one of them be odd? Why?

A:

Q: Find the number of subsets that can be formed from a set containing 10 elements.

A:

Q: How many triples of integers from 2 through 16 can be formed in which the integers are in increasing…

A: Any triple of integers (i, j, k) with 2 ≤ i ≤ j ≤ k ≤ 16 can berepresented as a string of 15 – 1…

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

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: 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: How many pairs of two distinct integers chosen from the set {1, 2, 3,..., 101} have a sum that is…

A: Since you have posted multiple questions and we can answer one question. So answer provided for the…

Q: How many ways are there to select three unordered elements from a set with five elements when…

A: Given: Number of elements = 5 Number of unordered elements = 3 If repetition is allowed,…

Q: How many pairs of integers (a,b) exists such that the product of a,b and HCF of (a,b) is 1080??

A:

Q: How many different integers must be selected to guarantee that at least 5 of them would have the…

A: Dividing any integer by 3, can have remainder 0, 1 or 2. Therefore, possibilities for remainder are…

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: How many integers in the set {1,2, ..., 10°} are divisible by 495 and not by 75?

A:

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

A:

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: Is the set O of all odd counting numbers (1, 3, 5… ) closed for multiplication? Why or why not?

A:

Q: Compute P5,2. (Enter an exact number.)

A: Obtain the value of P5,2.  The value of P5,2 is obtained below:   Permutation: A permutation is an…

Q: Let S={1,2,3,4,5,6,7,8,9,10,11}   How many subsets contain at least one odd number?

A:

Q: Assume that you have 8 dimes and 4 quarters (all distinct), and you select 3 coins. In how many…

A: The number of dimes is 8 and the quarters is 4.

Q: Compute the sum of the first 50 positive integers that are exactly divisible by 5.

A: To determine sum of the first 50 positive integers that are exactly divisible by 5.

Q: From the digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9, if five-digit numbers are to be formed and…

A: As per our guidelines we are supposed to answer only 3 sub-parts of any one question We have given…

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

A: Solution

Q: How many strings of 7 decimal digits are there, such that at least one of the digit is odd?

A: There are total 10 decimal digits, among which 5 are odd. Since repetition of digits is allowed,…

Q: How many pairs of two distinct integers chosen from the set {1, 2, 3, 101} have a sum that is even?

A:

Q: compute the sum of all integers between 60 and 110 that are exactly divisible by 7

A:

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: The total number of positive integers that can be formed from the digits 1,2,3 and 4 if no digit is…

A: Number of possible arrangement can be calculated either by combination or permutation methods.…

Q: How many integers between 1 and 10000(inclusive) are divisible by at one of 9, 10, 11?

A:

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

A: Given data: Set{1,2,3,4,5,6} To find: Three​-digit numbers  formed using elements from the set​.

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

A: In unordered sets, the order of the selection of the selection of items does not matter. So,…

Q: How many three-digit odd integers can be formed using the digits 0, 1, 4, 5, 6, 7, and 9 only once?

A: Permutation: A permutation of a set is defined as an arrangement of its members into a sequence or…

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: Show that if any 14 numbers are selected from the set S = {1,2,3,...,25}, [14] there are at least…

A: Note that the numbers in each of the first twelve sets sum to 26, and that there is no pair of…

Q: How many integers from 1 to 500 are divisible by at least one of  the numbers 3, 4, 5?

A: Given :  Integers from 1 to 500 Total numbers  = 500 So to calculate the number of divisors, we will…

Q: How many 5 digit positive integers are divisible by 5 and have at least one 6 as their digit?

A: A number is divisible by 5, if it has a 0 or 5 in the unit digit place. Let m choices to be made…

Q: Assume that you are to select digits from the set {2,5,6,8,9} to form a number between 20 and 900.…

A: Assume that you are to select digits from the set {2,5,6,8,9} to form a number between 20 and 900.…

Q: How many four-digit positive integers both start and end in even digits?

A:

Q: Find the number of ways 66 identical coins can be separated into three nonempty piles so that there…

A: GivenTotal number of identical coins =66Total number of piles =3 Each pile is on empty and the…

Statements and Logic

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