A Few Math Problems To Start With

Hello guys. I initially wanted to post a few problems from the IMO, but I decided to start off easy. So for today, try out these relatively simple problems.

1) At the beginning of math class every day, Mr. Smith selects students to write up homework problems on the board. These problems can be discussed as a class. There are 26 students in Mr. Smith’s math class, and he randomly selects with replacement a student to write up each of the first five problems. How many different ways can students be assigned to the problems?

2) A regular polygon is a polygon with equal angle measures and equal side lengths. A diagonal of a polygon connects two non-adjacent vertices. How many diagonals are there in a regular heptadecagon (17-sided polygon)?

3) George is traveling to New York City and has created a list of ten different possible sightseeing activities in which he is interested. He will be in New York City for three days, but will only have time for two different activities each day. How many different sightseeing plans can George create? (Assume each day is treated separately, and clearly George will not want to complete each activity more than once.)

Just to note, all these problems have to do with permutations/combinations. For tomorrow, I will introduce some concepts and go over these 3 questions.

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