8313: One Person Game
[Creator : ]
Description
There is a very simple and interesting one-person game. You have 3 dice, namely Die1, Die2 and Die3.
Die1 has $K_1$ faces. Die2 has $K_2$ faces. Die3 has $K_3$ faces. All the dice are fair dice, so the probability of rolling each value, $1$ to $K_1, K_2, K_3$ is exactly $1/K_1, 1/K_2$ and $1/K_3$.
You have a counter, and the game is played as follow:
Die1 has $K_1$ faces. Die2 has $K_2$ faces. Die3 has $K_3$ faces. All the dice are fair dice, so the probability of rolling each value, $1$ to $K_1, K_2, K_3$ is exactly $1/K_1, 1/K_2$ and $1/K_3$.
You have a counter, and the game is played as follow:
- Set the counter to $0$ at first.
- Roll the 3 dice simultaneously. If the up-facing number of Die1 is $a$, the up-facing number of Die2 is $b$ and the up-facing number of Die3 is $c$, set the counter to $0$. Otherwise, add the counter by the total value of the 3 up-facing numbers.
- If the counter's number is still not greater than n, go to step 2. Otherwise the game is ended.
Input
There are multiple test cases.
The first line of input is an integer $T\ (0 < T \leq 300)$ indicating the number of test cases.
Then T test cases follow.
Each test case is a line contains 7 non-negative integers $n, K_1, K_2, K_3, a, b, c\ (0 \leq n \leq 500,\ 1 < K_1, K_2, K_3 \leq 6,\ 1 \leq a \leq K_1,\ 1 \leq b \leq K_2,\ 1 \leq c \leq K_3)$.
The first line of input is an integer $T\ (0 < T \leq 300)$ indicating the number of test cases.
Then T test cases follow.
Each test case is a line contains 7 non-negative integers $n, K_1, K_2, K_3, a, b, c\ (0 \leq n \leq 500,\ 1 < K_1, K_2, K_3 \leq 6,\ 1 \leq a \leq K_1,\ 1 \leq b \leq K_2,\ 1 \leq c \leq K_3)$.
Output
For each test case, output the answer in a single line. A relative error of $10^{-8}$ will be accepted.
Sample 1 Input
2
0 2 2 2 1 1 1
0 6 6 6 1 1 1
Sample 1 Output
1.142857142857143
1.004651162790698