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

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Infinity Learn · Accepted Answer

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−$$1C2$$

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

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