P10908 [Lanqiao Cup 2024 National B] Number-Selection Probability

In an array, there are $a$ copies of $1$, $b$ copies of $2$, and $c$ copies of $3$. Let $P_{i,j}$ denote the probability that when two numbers are chosen at random from the array, one number is $i$ and the other number is $j$. For example, $P_{1,2} = \dfrac{ab}{C(a+b+c,2)}$, where $C(N, M)$ is a binomial coefficient, meaning the number of ways to choose $M$ elements from $N$ distinct elements. When $a=\text{?},b=\text{?},c=\text{?}$, it holds that $P_{1,2}=\dfrac{517}{2091},P_{2,3}=\dfrac{2632}{10455},P_{1,3}=\dfrac{308}{2091}$, and $a + b + c$ is minimal. It is guaranteed that the solution with minimal $a + b + c$ is unique. You need to submit a string in the format $a,b,c$. For example, if your computed result is $a = 12, b = 34, c = 56$, then the string you need to submit is `12,34,56`.

This is an output-only fill-in-the-blank problem. You only need to compute the result and submit it. The result of this problem is an integer. When submitting the answer, only fill in this integer; any extra content will result in no score.

This is an output-only fill-in-the-blank problem. You only need to compute the result and submit it. The result of this problem is an integer. When submitting the answer, only fill in this integer; any extra content will result in no score.

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