Probability · Question 10

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, (m, n) = 1, then m + n is equal to .

Answer

2477

Solution

Total ways = 40C3 = 9880

Let the G.P. be a, ar, ar2 with r = pq > 1, (p, q) = 1.

For all terms to be integers, a = kq2; terms are kq2, kpq, kp2 and we need kp2 40.

Number of such G.P.s for each p is (number of q < p coprime to p) [ 40p2 ]:

p = 2 (q = 1): 1 10 = 10

p = 3 (q = 1, 2): 2 4 = 8

p = 4 (q = 1, 3): 2 2 = 4

p = 5 (q = 1, 2, 3, 4): 4 1 = 4

p = 6 (q = 1, 5): 2 1 = 2

p 7 p2 > 40: none.

Favourable = 10 + 8 + 4 + 4 + 2 = 28

Probability = 289880 = 72470 (2470 = 2 5 13 19, so already in lowest terms)

m + n = 7 + 2470 = 2477

Answer: 2477