Three students were to be chosen from 8 candidates to be school prefects.

Question

Three students were to be chosen from 8 candidates to be school prefects. Lauren calculated-correctly-  that it's possible to choose 3 people from 8 in 56 ways. Which of the following statements must also be true, based on Lauren's calculation"?

 

a. You can choose 6 people from 8 in 112 ways

 

b. You can choose 6 people from 16 in 112 ways

 

c. You can choose 4 people from 8 in 75 ways

 

d. You can choose 5 people from 8 in 56 ways

Answer 1

8!/3!(8-3)!

8!/3!5!

(8*7*6)/(3*2)

8*7

56

 

8!/5!(8-5)!

8!/5!3! same as above

Answer 2

When 3 are chosen, it's like putting those 3 in one box.  Imagine putting the other 5 in a different box.

 

Every time you choose 3, it completely determines what the other 5 are.  That is, for each choice of 3 there is a one and only one specific choice of 5 that go in the other box.  So if there are 56 different groups of 3 you could put in the first box, there are 56 different groups of 5 that you put in the other box.  There is no way to choose some other group of 5.

 

So there are just as many ways to choose 3 and put them in one box as there are to choose 5 and put them in the other box.

 

Therefore the answer is d.