To get to a friendSolutionThe paths can be described by word

To get to a friend

Solution

The paths can be described by \"words\" formed from the letters D(for down) and R(for right.) We are asked to determine the number of D\'s if the total number of permutations is 66 and the number of right turns is 2.

Let x represent the total number of letters in the \"words\". (E.g. if x=3 then we can have RRD,RDR,DRR as the only possible paths since there are 2 R\'s with R=right turn and D=down turn.)

Then the number of permutations, 66 , can be represented as:

[(x!)/((x-2)!2!)=66] (x! represents the total number of words -- we divide to find the number of distinguishable permutations.)

[(x!)/((x-2)!2!)=66]

[(x(x-1)(x-2)!)/((x-2)!2)=66]

[x(x-1)=132]

[x^2-x-132=0]

[(x-12)(x+11)=0]

[x=12,x=-11] The second answer does not make sense in context, so x=12.

Thus the length of the words is 12; there are 2 R\'s so there must be 10 D\'s.