Feller shoe-matching problem This problem have been taken from the book' An Introduction to Probability Theory and Its Applications' by Williams Feller(1906-1970)
Note:- Assume in each case that all possible arrangements have the same probability.
Ten pair of shoes are in a closet. Four shoes are selected at random.Find the probability that there will be at least one pair among the four shoes selected
Solution:-
Answer provided by the author is $\frac{\binom{55}{2}}{\binom{20}{4}}=\frac{1485}{4845}=\frac{99}{323}$
we want to find the probability that there will be at least one pair of shoes among the four shoes selected which is equal to the probability that remains after deducting the probability of no pairs of shoes among the four shoes selected from the total probability.
Let us calculate the probability of picking 1st,2nd,3rd and 4th shoes so that there are no pairs. 
The probability of 1st shoes 20/20
2nd shoes 18/19, 3rd shoes 16/18 and 4th shoes 14/17.
If we deduct the product of these result from the total probability, we get the our desired result.i-e $1-\frac{20*18*16*14}{20*19*18*17}=\frac{99}{323}$
combination
 A: Comment: Here is a simulation in R of a million draws of four shoes from the closet. In the closet each pair has its own number from 1 to 10. If my draw
results in exactly four uniquely different numbers, I have no pairs. Otherwise,
I have at least one pair. The vector nr.unique has a million entries, each
of which can be a number from 2 (I drew 2 pairs of shoes) through 4 (no pairs).
The vector nr.uniq < 4 is a 'logical' vector containing TRUEs and FALSEs.
Its mean is its proportion of TRUEs.
With a million draws the margin of simulation error should be less than 0.001
in 95% of such simulations. Results agree with Feller's answer.
set.seed(921)
closet = rep(1:10, 2)
nr.uniq = replicate( 10^6, length(unique(sample(closet,4))) )
mean(nr.uniq < 4);  99/323
[1] 0.306618      # simulated result
[1] 0.3065015     # Feller's answer

table(nr.uniq)    # tally counts
nr.uniq
     2      3      4 
  9377 297241 693382 
2*sd(nr.uniq < 4)/sqrt(10^6)
[1] 0.0009221792  # aprx 95% margin of sim err

