What is the probability that on a given day, the number of half gallon containers provided is enough?

Question

In a grocery store 400 customers shop every day. The number of half gallons of nonfat milk bought by a randomly selected customer is a random variable X having P(X=0)=0.3, P(X=1)=0.5, and P(X=2)=0.2. Assume buying behaviors of different customers are independent. The grocer requests 390 half gallon containers per day from the ditributor. What is the probability that on a given day, that is enough?

Answer 1

Using the the multinomial theorem

$$\frac{400!}{k!\ l!\ (400-k-l)!}(0.5)^k(0.2)^l(0.3)^{400-k-l}$$

is the probability of any combination. You want to sum all $(k,l)$ pairs such that $k+2l\leq 390$

This will be difficult to actually evaluate, you might want to use approximations.