8. What is the minimum number of students, each of whom comes from one of the 5 boroughs, who must be enrolled in a class to guarantee that there are at least 15 who come from the same borough? Justify.
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- The number of 5-element subsets from a set containing n elements is equal to the number of 6-element subsets from the same set. What is the value of n? (Hint: the order in which the elements for the subsets are chosen is not important.)What is the minimum number of students, each of whom comes from one of the 17 regions, who must be enrolled in a university to guarantee that there are at least 100 who come from the same region?Solve using " Pigeonhole Principle "(a) There are 7 different time periods in a day during which classes at BRAC University can bescheduled. If there are 597 classes of different courses held per day, what is the minimumnumber of classrooms that will be needed? (b) What is the minimum number of students, each of whom comes from one of the 64 districtsof Bangladesh, which must be enrolled in BRAC University to guarantee that there are at least200 who come from the same district?
- Solve these problems using "Pigeonhole Principle".1) How many numbers must be selected from the set {20,30,40,50,60,70,80} to guarantee thatat least one pair of these numbers add up to 110? 2) Three parties are running for Bangladesh’s upcoming gubernatorial election. There are5,00,500 people who vote. What is the minimum number of votes needed for any party to winthe election?What is the minimum number of balls that we need to draw to ensure that we get at least 3 balls which end with the same digit (i.e., all three balls have the same digit in their ones place)?A bag contains 10 red marbles, 10 white marbles, and 10 blue marbles. What is the minimum number of marbles that need to be picked to ensure that at least 8 marbles have the same color?
- a drawer contains 17 Black socks, 17 White socks, 17 Blue socks and 17 brown socks. What is the minimum number of sockas that need to be pulled to get a matching pair?What is the minimum number of students required in a Mathematics class test to be sure that at least 9 will receive the same grade, if there are 5 possible grades A, B, C, D and E.1. A teacher administers a multiple-choice quiz with four questions and four possible answers: a, b, c, and d. What is the bare minimum of students required in the class to ensure that at least three answer sheets are identical?2. What is the minimum number of students from each of the 15 regions who must be enrolled in a university to ensure that there are at least 52 from the same region?
- The International Telecommunications Union (ITU) specifies that a telephone number must consist of a country code with between 1 and 3 digits, except that the code 0 is not available for use as a country code, followed by a number with at most 15 digits. How many available possible telephone numbers are there that satisfy these restrictions?Twenty-five friends play golf in 5 groups of 5. Show that 6 is the minimum number of rounds they must play so that each person plays with everyone at least one time. Give such a schedule