Permutations and Combinations

Question

An odd number of stones lie along a straight path, the distance between any two consecutive stones being 10 m. The stones are to the collected at the place where the middle stone lies. A man can carry only one stone at a time. He starts carrying the stones beginning from the extreme stone. If he covers a path of 3 km, how many stones are there ?

Answer

Let the total number of stones be 2n + 1
∴ There are n stones on each side of the middle stone. Let the man starts collecting stones from the extreme left stone. The distance covered by man to bring the extreme left stone to the middle stone = 10n metres. The distance covered by man to bring the (n - 1)th stone on left to the middle stone = 2 $cross times$ 10 (n - 1) metres.
∴ The total distance covered by the man to bring all the stones from left to the middle stone
= 10n + 20(n - 1) + 20(n - 2) + ......... + 20
Similarly, the total distance covered by the man to bring all the stones from the right to the middle stone
= 20 + 20 (n - 1) + ......... + 20
∴ Total distance covered by the man to bring all the stones to the middle stone
= 30n + 40 (n - 1) + 40 (n - 2) +........ + 40 (1)
= 40 [n + (n - 1) + (n - 2) + ....... + 1] - 10n = $40 sum from straight k equals 1 to straight n of straight k minus 10 straight n$
$equals fraction numerator 40 straight n left parenthesis straight n plus 1 right parenthesis over denominator 2 end fraction minus 10 straight n$
But the distance covered is 3 km or 3000 metres.
∴ $fraction numerator 40 straight n left parenthesis straight n plus 1 right parenthesis over denominator 2 end fraction minus 10 straight n space equals space 3000 space space space space rightwards double arrow space space space 20 straight n squared plus 20 straight n minus 10 straight n space equals space 3000$
$rightwards double arrow$ $20 straight n squared plus 10 straight n minus 3000 space equals space 0$ $rightwards double arrow$ $2 straight n squared plus straight n minus 300 space equals space 0$
$rightwards double arrow$ $space space space space space straight n equals space fraction numerator negative 1 plus-or-minus square root of 1 plus 4 left parenthesis 2 right parenthesis left parenthesis 300 right parenthesis end root over denominator 2 left parenthesis 2 right parenthesis end fraction space equals space fraction numerator negative 1 plus-or-minus square root of 2401 over denominator 4 end fraction space equals space fraction numerator negative 1 plus-or-minus 49 over denominator 4 end fraction space equals space 48 over 4 comma space fraction numerator negative 50 over denominator 4 end fraction$
$rightwards double arrow$ n = 12
∴ Number of stones = 2(12) + 1 = 25