A car is parked among N cars standing in a row, but not at either end. On his return, the owner finds that exactly ''r" of the N places are still occupied. The probability that the places neighboring his car are empty is

Since there are r cars in l/ places, total number of selection of places out of N - 1 places for r - 1 cars (except the owner's car) is N−1Cr−1=(N−1)!(r−1)!(N−r)!If neighboring places are empty, then r - 1 cars must be parked in N - 3 places. So, the favorable number of cases is N−3Cr−1=(N−3)!(r−1)!(N−r−2)!Therefore, the required probability is=(N−3)!(r−1)!(N−r−2)!×(r−1)!(N−r)!(N−1)!=(N−r)(N−r−1)(N−1)(N−2)= N−rC2 N−1C2