6 friends go out for dinner and order from a menu of 8 different sandwiches. What is the probability that at least 2 of the group order the same sandwich?

Question

Factorials

Answer 1

P(at least 2 order same) = 1 - P(all order different)  

 

To find P(all order different), let 

 

NT = total number of possible orders without regard to same or different 

ND = total number of orders that are all different  

 

To find NT, note that each of the 6 friends has 8 choices for their sandwich, so NT = 8⁶ = 262144  

 

To find ND, imagine the waiter asking each friend in order which sandwich they want.  Friend 1 can choose any of the 8 sandwiches.  To be different, friend 2 can only choose from 7 of the remaining sandwiches.  Friend 3 will only have 6 choices, friend 4 will have 5 choices, friend 5 will have 4 choices, and friend 6 will only have 3 choices.  So  

ND = 8*7*6*5*4*3 = 20160  

 

P(all different) = ND / NT = 20160 / 262144 = 315 / 4096 ~ 0.0769  

 

So P(at least 2 same) = 1 - (315/4096) = 3781/4096 ~ 0.9231