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Math Help - Arranging letters in a word, cant understand this

  1. #1
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    Arranging letters in a word, cant understand this

    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?
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  2. #2
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    Re: Arranging letters in a word, cant understand this

    Quote Originally Posted by nukenuts View Post
    Find the number of different arrangements of the letters in the place name WOLLONGONG that do not contain two letter L's consecutively.
    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.
    Look at _W_N_N_G_G_O_O_O_. Do you see nine places to put the L's?
    Choose two of them. The L's will be separated.
    Now the separators can be arranged in \frac{8!}{(2!)^2(3!)} ways.
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  3. #3
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    Re: Arranging letters in a word, cant understand this

    Cheers mate.
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