I am trying to calculate the probability of at least 2 people sharing a birthday in a group of 4 people. I understand that calculating it as 1-P(no shared birthdays) is simpler, but I would like to understand the counting method by doing it directly.
My attempt for $n=4$ is
P = P(2 people) + P(3 people) + P(4 people) = $\frac{1}{365}\binom{4}{2}+\frac{1}{365^2}\binom{4}{3}+\frac{1}{365^3}\binom{4}{4}=0.0164$...
but this does not match up with
P = $1-\frac{364}{365}\frac{363}{365}\frac{362}{365}=0.163...$
What am I doing wrong in the direct calculation?