Subjective Type

If $$N$$ is the number of ways in which $$3$$ distinct numbers can be selected from the set $$\left \{3^{1}, 3^{2}, 3^{3}, ... 3^{10}\right \}$$ so that they form a G.P. then the value of $$N/5$$ is.

Solution

If three numbers are in G.P., then their exponent must be in A.P.
If a,b,c are selected number in G.P., then the exponents of a and c both are either odd or both even, or otherwise exponent b will not be integer.
Now two odd exponent (from 1,2,3,...,10) can be selected in 5C2 ways and two even exponent can be selected in 5C2 ways.
Hence number of G.P.'s are 25C2=20.

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