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

Q: How many integers between 100 and 400 contain the digot 2?

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Q: How many positive integers less than 10000 are there which contain at least one 7 or at least one 9…

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Q: Suppose we define a function f(x) by f(x) = a1 sin(x) + a2 sin(2x) +· + an sin(nx), .. for constants…

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Q: How many integers from 1 to 700 (inclusive) are divisible by the factors of 70 greater than 1?

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Q: Produce the least positive number that is divisible by​ 2, 3,​ 4, 5,​ 6, 7,​ 8, 9,​ 10, and 11.…

A: Given Produce the least positive number that is divisible 2, 3,​ 4, 5,​ 6, 7,​ 8, 9,​ 10, 11.…

Q: Explain why, in any set of 11 positive integers, some two integers must have the same I's digit.

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Q: Suppose 35 cars start at a car race. In how many ways can the top 3 cars finish the race?

A: The information provided in the question are as follows :-There are  cars at a car race.

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: 5. (12 points) Find how many positive integers between 1 and 99, inclusive, have the following…

A: The objective of this question is to find the number of positive integers between 1 and 99 that are…

Q: During a year ending July 31 of that year, there were approximately 6.2 million sales of existing…

A: Number of  homes sold in West = 1.4 million Number of homes sold in United States = 560000Number of…

Q: exp(ix())= 4k+1 2

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Q: Texas is starting a lottery game. Play- ers use a machine that randomly selects 7 distinct numbers…

A: To calculate the probability that Texas will lose money, we need to determine the expected value of…

Q: How many 5–digit (decimal) integers are there with none of the digits repeated?

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Q: How many numbers between 1 and 10000 have the digits 1,3, and 5 such that each digit appears exactly…

A: Here Numbers between 1 and 10,000 have digits 1,3,5 and each appears exactly once only

Q: n any positive integer n, there exist n consecutive composite integers

A: Given that given any positive integer n, show that there exist n consecutive composite integers.

Q: "find A2, A−2, and A−k (where k is any integer) by inspection."

A: NOTE: Refresh your page if you can't see any equations. . use the diagonal matrix formula

Q: How many distinct four-digit integers can one make from the digits 1,3,3,7,7, and 8?

A: How many distinct four-digit integers can one make from the digits 1,3,3,7,7, and 8?

Q: - Let a, b, n be positive integers. Prove that if a = b (mod n), then (a, n) = (b, n).

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Q: How many integers in the set {1,2,3,4, ..., 200} is divisible by at least one of the primes 2, 5 and…

A: Just use inclusion-exclusion principle for three sets.

Q: show that every integer greater than 11 is the sum of two composite integers

A: We need to prove that  every integer greater than 11 is the sum of two composite integers.

Q: You are given 3  ropes and   a lighter.The  time required  to burn through rope 1  with a  single…

A: You are given 3  ropes and   a lighter. The  time required  to burn through rope 1  with a  single…

Q: If n is a positive integer, is there always a prime between n and 2n?

A: given that n is a positive integer. we have to show that there is always a prime number between n…

Q: How many integers between 100 and 400 contain the digit 2? There are integers between 100 and 400…

A: Numbers between 200 and 299 contains 2 So total integers = 100 Numbers between 120 – 129 contains 2…

Q: 3. How many 5-digit positive integers are there in which there are no repeated digits and all digits…

A: 3 We have to find the number of 5-digit positive integers are there in which there are no repeated…

Q: MK18. Show that in any set of 9 integers chosen from the set {2,3, 4, ... , 21, 22}, there are at…

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Q: How large a subset of the integers from 1 to 50 must be chosen to guarantee that one of the numbers…

A: We have to determine How large a subset of integers from 1 to 50 must be chosen to guarantee that…

Q: prove that the sum of four consecutive integers is even.

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Q: How many six-number combinations can be generated from the numbers from 1 to 42 inclusive, without…

A: Given information: A 6 number combination is to be generated. The possible numbers are selected…

Q: How many corresponding integers are made up of 1443 digits taken from (1-2-3-4-5-6) provided that no…

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Q: Consider all positive integers with seven different digits. (Note that zero cannot be the first…

A: Given that we have to find all positive integers with seven different digits which satisfy the given…