Consider a population $N$.
If all pairs of finger print matchings were independent, the number that we want to calculate would just be $p^{^NC_2}$.
But they are obviously not independent since if we know that Ram's fingerprint doesn't match with Laxman's and Bharat's, then it changes the probability of finding a match between Laxman's and Bharat's fingerprints.
How should I go about calculating the probability that I have asked in the question?
On
I assume you mean the fraction in 1000.
The actually probablility can be googled. It is 1 in 64,000,000. But let's call it P in case you need to use the method in a different context. Well. First let's make 1/P x/1000. The simplest idea is to find out P/1000 and apply that answer to the numerator (1) So:
(1÷(p/1000))/(p÷(p/1000))...
If we substitute P with 64,000,000 we end up with 0.000015625/1000
I'm not sure about my earlier answer, but I have a better one:
The probability of a person's fingerprint not match with someone else is:
63,999,999/64,000,000 which is P
Great, so now we need to know if N people have matching fingerprint. If we a add a third person the probability that it doesn't match one of the earlier people is 63,999,998/64,000,000 which is two probabilities. For a fourth it would be 63,999,997/64,000,000. Multiplying these probabilities together give us:
$\frac{63999999}{64000000}\cdot\frac{63999998}{64000000}\cdot\frac{63999997}{64000000}...n\ times$
This can be written as a factorial:
$\frac{\frac{\left(63999999!\right)}{\left(64000000-n\right)!}}{64,000,000^{n}}=\frac{63999999!}{\left(64000000-n\right)!\cdot64,000,000^{n}}$
Note if N is 64,000,001, you would reach a probability of 0 fro the integer values, indicating it's impossible for none of the finger prints to be alike. Unless my previous answer was what you were looking for, sorry you felt the need to tick that one.