The number of possible poker hands drawn from a standard 5-card deck is ${52 \choose 5}$. This is sampling without replacement where order does not matter, e.g., A-K-Q-J-10
is the same hand as 10-J-A-K-Q
.
In a group of 5 individuals the number of possible sequences of birthdays (using the standard rules of the Birthday Problem) is $365^5$. This is sampling with replacement where order does matter, e.g., 1-7-23-23-314
is different than 23-7-23-314-1
. I can't seem to develop any intuition for why order should matter in the Birthday Problem, i.e., why isn't the number of possible sequences ${365+5-1 \choose 5}$?
Can someone help me out and explain why order matters?