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Question

Out of 31 numbers 1,2,3,...,31, three numbers are chosen at random, without replacement. Let p denote the probability that the selected numbers are in A.P. find 899p.

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Solution

Any three numbers out of 31 van be chosen in 31C3 ways.
For the favourable choice if the chosen numbers are a,b and c, then a+c2=b.
Obviously either both a and c are even or both are odd and then b is fixed.
Hence for favourable choice we have to chose two numbers from 1 to 31 in
16C2+15C2 ways.
Hence required probability is 16C2+15C231C3=8×15+15×75×31×29=45899

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