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This problem trains for: SAT-I, AMC-8, GMAT.
A dog has buried bones in 10 caches as follows: 5, 10, 15,... bones. What is the probability that by opening two caches at random the dog finds an average of 20 bones per cache?
We have to count in how many ways the dog can find 40 bones in total by opening two caches:
Note that this sum cannot be realized by adding up 20 and 20 since there are no two caches with the same number of bones inside. Same discussion for zero bones.
Since each of these sums can be realized in two ways (e.g. (3,35) is the same as (35,5)) there are a total of 6 ways to accomplish the desired sum.
The total number of sums that can result from opening two caches at random is a combination since the order of opening does not matter:
The probability is given by the ratio between the number of favorable outcomes and the total number of possible outcomes: