Find the number of different arrangements of the letters in the place name WOLLONGONG that do not contain two letter L's consecutively.
I know of one method of doing this problem -
1. Find total number of arrangements of this word = 75600
2. Find number of ways it can be arranged with L's consecutively = 15120
3. Subtract 2. from 1. = 60480
But from a solution Ive been given it is done another way and I dont understand this -
1. There are 1680 ways to arrange the 8 non-Ls ----> 8 Choose 1,2,2,3
2. Then there are 9 possible locations for the non-consecutive Ls, which can be chosen in 9 Choose 2 = 36 ways yielding 60480 of these arrangements in total.
The bolded part is what I dont understand. How are there 9 possible locations for the non-consecutive Ls...? It seems to me that we have a choice of 10 positions for the first L and then 8 for the second L (as they cant be beside each other). Then divide this by two to take account of double counting. But this does not give the correct answer.
Anyone understand how the bolded part works?