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Permutation and Combination

1. A cube has all of its sides painted in red and then is cut into 216 cubes of the same size. All cubes are placed in an urn and are thoroughly mixed, so that the probability of being randomly picked from the urn is the same for all cubes. What is the probability that a randomly picked cube has at least one of its sides painted red?

2. There are ten raffle tickets, two of which are winners. Find the probability that in a sample of 6 tickets there will be no more than one winning ticket.

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Solution Preview

1. Let us consider that each of the 216 cubes of the same size is of dimension (x*x*x), so the original cube that is cut into these 216 cubes, must be of size (6x*6x*6x).

Largest sub-cube contained inside this big cube that will not touch the outer cube on any side, will of dimension (6x-x-x)*(6x-x-x)*(6x-x-x) i.e. (4x*4x*4x); any small cube of dimension (x*x*x) created out of this sub-cube will not have any of it's side painted red.

So, total number of cubes that will not have any side painted red = 4*4*4 = 64

P(at least one side painted red) ...

Solution Summary

Solution gives enough explanations to enable the student solve similar problems easily. More than one ways to get the solution are discussed in the answer to second question.