Question

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 $ \frac{m}{n} $, where $ \gcd(m, n) = 1 $, then $ m + n $ is equal to:

Hint:

When selecting numbers in a geometric progression, consider each possible common ratio and check for conditions on \( a \).

Answer 1

Correct Answer: 4

Let the numbers selected be \( a, ar, ar^2 \), where \( a \) and \( r \) are in \( \mathbb{N} \). 
We evaluate for different values of \( r \): For \( r = 2 \), we calculate possible values of \( a \), and similarly for other values of \( r \). 
After summing the probabilities for each possible case, we find: \[ \text{Total} = 28\]  
Thus, \(\ m+n = 13. \)