With solutions please I\'m unable to calculate them with my calculator 1. Suppos

Question

With solutions please I'm unable to calculate them with my calculator

1. Suppose 100 students are in your school. The school’s principal wants to take photos of everybody in different ways so that every student thinks they’ve been treated fairly.

a) The principal’s first idea is to take photos of every possible group of 10 students, lined up in every possible way. How many photos are going to need to be taken in this situation?

b) The principal didn’t realize it would be that many photos. Instead she has decided that she wants photos of every possible group of 10 students, but doesn’t care how they’re lined up. How many photos are going to need to be taken in this situation?

c) The principal didn’t realize it would be that many photos. Instead she has decided that she just wants to divide the 100 students into groups of 10, and take one photo of each group. So, no student is in more than one photo, and she doesn’t care how they’re lined up. How many photos are going to need to be taken in this situation?

d) The principal doesn’t think that’s a lot of photos, so she has a new idea that includes three of the school’s teachers. She still wants to divide the 100 students into groups of 10. With each student group, they will take a picture with every possible way to choose two of the three teachers, with one on the left and the other on the right (so, the order matters). How many photos are going to need to be taken in this situation?

Explanation / Answer

(A) There are (10010)(10010) different groups of 1010students. Each group of 1010 students can be arranged in 10!10! different ways: 1010ways to choose the first student, 99 ways to choose the second student, ... , 11 way to choose the last student. Therefore the total number of pictures is 10!10!(10010)(10010)=100!(10010)!

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