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Solution


Q.

Three distinct numbers are selected randomly from the set {1, 2, 3, ..., 40}. If the probability, that the selected numbers are in an increasing G.P., is mn, gcd (m, n) = 1, then m + n is __________.          [2025]

Ans.

(2477)

When r∈N

Common ratio Last triplet Total number of G.P. formed
r = 2 10, 20, 40 10
r = 3 4, 12, 36 4
r = 4 2, 8, 32 2
r = 5 1, 5, 25 1
r = 6 1, 6, 36 1
  Total 18

When r∉N (also possible)

Common ratio Last triplet Total number of G.P. formed
r = 3/2 16, 24, 36 4
r = 5/2 4, 10, 25 1
r = 4/3 18, 24, 32 2
r = 5/3 9, 15, 25 1
r = 5/4 16, 20, 25 1
r = 6/5 25, 30, 36 1
  Total 10

Total number of choices =C340=9880

Required probability =289880=72470=mn

∴  m = 7 and n = 2470

Hence, m + n = 7 + 2470 = 2477.