Suppose that in a particular Finite course, 4/7 of students plan on majoring in math, 25/42 plan on majoring in communications, and 2/7 plan on double majoring in math and communications. How many want to major in exactly one of communications or math?

A report in the college newsletter used the registrar's database for all the students in the school to look at majors and classes being taken this semester. The report includes the sentences:

3. A company checks its managers and find that 70 of them have an undergraduate degree in Business, 30 have an undergraduate degree in business and a Masters in Business, and 45 have only A Masters in Business. 20 others have neither degree. How many managers are in this company?

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 burger chain wants to send codes for a free small drink to the members of its kids club. The coupon code will have one letter, followed by two digits, followed by two letters. The letters  R ,  S ,  T , and  U  and the digit  3  will not be used. So, there are  22  letters and  9  digits that will be used. Assume that the letters and digits can be repeated. How many such coupon codes can be generated?

Suppose you have 11 books and you randomly pick 3 to take on vacation with you. How many ways can you do this?

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.