Interesting Probability Question - Birthday Problem Variation

The pickle factory can produce arbitrarily many pickles by hiring only people born on a particular day.

To make a problem in the Birthday Problem tradition, we will assume that birthdays of pickle makers are independent uniformly distributed, and that a year has $365$ days. . Suppose we hire $n$ people.

For $i=1$ to $365$, let $Y_i=1$ if no one has a birthday on Day $i$, and let $Y_i=0$ otherwise. Then the pickle production on Day $i$ is $10nY_i$. The yearly production is $\sum_1^{365}10nY_i$, and the expected yearly production, by the linearity of expectation, is $\sum_1^{365}10nE(Y_i)$.

The probability no one has a birthday on Day $i$ is $(364/365)^n$. So we want to maximize $$3650n(364/365)^n.$$ This is a standard calculus problem, except that we will have to produce an integer answer. Use the calculus to maximize $te^{-kt}$ where $k=\ln(365/364)$.

Remark: We used the linearity of expectation to sidestep the more complicated problem of finding the distribution of the number of pickle jars produced. That approach leaves us with a complicated expression for the expectation. Indicator random variables such as our $Y_i$ can be very useful.

I know this has already been answered mathematically, but for what it's worth, here is a histogram of average values gotten from random trials.

You can clearly see a maximum somewhere around 365 employees.