# A discrete mathematics problem by Soumava Pal

$$A$$, $$B$$ and $$C$$ take turns rolling a fair die (with equal probability of it showing any of $$\{1,2,3,4,5,6\}$$, and the as soon as one of them gets a six, the game ends and he wins.

$$A$$ gets the first chance to roll the die ,followed by $$B$$, and then $$C$$, followed by $$A$$ again if none of them gets a six.

Let the probability of $$A$$ winning the game be $$\dfrac{a}{b}$$, that of $$B$$ be $$\dfrac{c}{d}$$ and that of $$C$$ winning the game be $$\dfrac{e}{f}$$, with $$a,b,c,d,e,f$$ being positive integers, with $$\gcd(a,b)=\gcd(c,d)=\gcd(e,f)=1$$.

Find $$a+b+c+d+e+f$$.

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