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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?

I was thinking the way to do this would be:

Number of ways for every possible group of 10 students $\cdot$ Number of ways to order each student in the group= Answer

I got so far:

? * $\left(\frac{10!}{(10-10)!}\right)$

I cant seem to find a way to calculate how to find the number of ways for every possible group of 10 students. Some help would be appreciated

Ike Casimir
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2 Answers2

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There are ${100}\choose{10}$ different groups of $10$ students. Each group of $10$ students can be arranged in $10!$ different ways: $10$ ways to choose the first student, $9$ ways to choose the second student, ... , $1$ way to choose the last student. Therefore the total number of pictures is $10!$${100}\choose{10}$=$\frac{100!}{(100-10)!}$

Dan
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The number you are looking for is

$$\displaystyle {100 \choose 10} = \dfrac {100!}{10!(100-10)!}$$

But I would argue "group" means an unordered collection, so I think the answer should just be $100 \choose 10$.

Ovi
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