Suppose that there are 27 students in discrete mathematics class. Show that the class must have at least 14 male students or at least 14 female students.

4) a. If 35! = x, then find 33! in the term of x (without using calculator). b. Show that if 5000 boys registered for driving license during the last month at driving school in Muscat, then at least 167 of them were registered on the same day of the month.

By the Pigeonhole Principle Show that if there are 60 students in a class, then at least two have last names that begin with the same letter.

If you have 12 boxes then how many balls do you have to put randomly into the boxes in order to guarantee that some box will contain at least 4 balls?

A group of 650 tourists includes 248 who have visited Germany, 115 who have visited Holland,and 319 who have visited neither Germany nor Holland. Use the Addition Principle to computehow many have visited both Germany and Holland.

We want to divide 12 children into any number of playgroups. However, in order to avoid some predictable disagreements, we want Adam and Brenda to be in different playgroups. How many ways are there to do this? (Playgroups have at least one child)

A card game contains 52 cards where 13 are of hearts, 13 are spades, 13 are clubs and 13 are diamonds. What is the minimum amount of cards that we would need to pull to get at least 6 cards of the same symbol?

Find the number of ways to select 12 objects with repetition from 5 distinct types of objects with at least 1 objects of each type and at most 4 objects of each type.

2. (a) A sack contains 50 marbles of 4 different colors. Explain why there are at least 13 marbles of the same color. (b) If exactly 8 of the marbles are red, explain why there are at least 14 of the same color.

5. A box contains 9 Red and 8 Green balls. In how many ways can 4 balls be drawn such that (a) they are all Red and (b) 2 Red and 2 Green balls?