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:
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, \(\ m+n = 13. \)
Given below are two statements I and II.
Statement I: Dumas method is used for estimation of "Nitrogen" in an organic compound.
Statement II: Dumas method involves the formation of ammonium sulfate by heating the organic compound with concentrated H\(_2\)SO\(_4\). In the light of the above statements, choose the correct answer from the options given below:
The motion of an airplane is represented by the velocity-time graph as shown below. The distance covered by the airplane in the first 30.5 seconds is km.