408782: GYM103316 I Cabbage Dan

Cabbage Dan hears that Avatar Tang will be far away in the Fire Empire land for the day and realizes that it is finally safe for him to set out his cabbage stall for sales. Due to his long period of inactivity, lots of folks are lined up to get one of his world-famous cabbages. As $$$n + m$$$ people gather around the stall, he realizes that $$$n$$$ of them each brought exactly one copper coin, and the remaining $$$m$$$ of them brought exactly one silver coin and no other currency. In the Earth Empire, two copper coins are equivalent to one silver coin and Dan charges a single copper coin for a single cabbage. All his customers are extremely frugal so they will only buy a single cabbage even though they might be able to buy more.

Dan starts the day with $$$k$$$ copper coins but he has no idea what order the $$$n + m$$$ customers will actually come to pay (in fact the customers will come in random order with all orderings being equally likely). Dan doesn't want to run into the embarrassing situation where a customer pays him in a silver coin but he doesn't have a copper coin to give back in change. Thus, he asks you to compute the probability that he can help all his customers and give appropriate change to everyone over the course of the day.

The first and only line of input will contain $$$3$$$ space-separated integers $$$n$$$, $$$m$$$, and $$$k$$$ ($$$0 \leq n, m \leq 10^{5}$$$ and $$$0 \leq k \leq 20$$$) where $$$n$$$ is the number of people paying with a copper coin, $$$m$$$ is the number of people paying with a silver coin, and $$$k$$$ is the number of copper coins Cabbage Dan begins the day with.

The probability that Cabbage Dan is able to service all customers and give change to everyone that needs it, answers will be accepted if they are within $$$10^{-6}$$$ of the correct answer.

4 2 1

0.9333333333

5 3 1

0.8571428571

0 2 2

1