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A cube is painted on all its faces. It is then cut into 64 smaller identical cubes, which are then thoroughly. What is the probability that 2 randomly chosen smaller cubes have exactly 2 coloured faces each?

I have a doubt , there will be 24 identical cubes whose 2 faces will be painted, 8 identical cubes with 3 faces painted, 24 identical cubes with one face painted and 8 identical cubes with no face painted. Now total cases to pick 2 smaller cubes should be 4C2 not 64C2 . correct me if I am wrong

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    Would that lead you to claim that the chance of picking any given color is $\frac 14$? This is not correct. Put 10 blue marbles and one red marble in a hat, draw 20 times with replacement, and see if you get around 10 reds.2012-05-05

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We can also work directly with the probabilities. Pick the cubes a little cube at random, then pick another. The probability that the first cube picked has exactly $2$ red faces is $\frac{24}{64}$. Given that the first cube picked had exactly $2$ red faces, the probability that the second cube picked has exactly $2$ red faces is $\frac{23}{63}$. Thus the probability we get two cubes with exactly $2$ red faces is $\frac{24}{64}\cdot\frac{23}{63}$.

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    In his comment, the OP suggested that it "should be one."2012-05-05
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The problem is that picking the different types of cube has different probabilities. It is easier to consider the 64 cubes distinguishable (number them) and then figure out how many of the choices meet your requirement. In this case, it also deals with the fact that picking a 2 face cube the first time changes the probability that the second pick will be a 2 face cube. In fact, my space would be $64 \cdot 63$, not $64 \choose 2$. How many of those are both 2 face cubes?

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    How you are making them distinguishable when I said Identical2012-05-05