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Math Help - How many ways can the word "Think" be arranged?

  1. #1
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    How many ways can the word "Think" be arranged?

    Hi,

    I can't seem to figure this one out. How many ways can the work "THINK" be arranged so that the "T" and the "H" are separated by at least one letter?

    I'm trying 5x3x3x2= 90 ways but that isn't working. I think I need to use the rule of sum but I'm not quite sure how. The answer is suppose to be 72. How?

    Thanks!

    P.S. I have another question I'm unable to do: A locksmith has ten types of blanks for keys. Each blank has five different cutting positions and three different at each position, except for the first position, which only has two depths. How many different keys are possible with these blanks?

    I'm tried 10x5x3= 150-2 = 148 but that answer is actually 1620. How is this?

    Can someone please explain to me how to complete both of these questions?
    Last edited by demo1; September 10th 2009 at 01:12 PM.
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  2. #2
    MHF Contributor alexmahone's Avatar
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    Quote Originally Posted by demo1 View Post
    Hi,

    I can't seem to figure this one out. How many ways can the work "THINK" be arranged so that the "T" and the "H" are separated by at least one letter?

    I'm trying 5x3x3x2= 90 ways but that isn't working. I think I need to use the rule of sum but I'm not quite sure how. The answer is suppose to be 72. How?

    Thanks!

    Total no of words that can be formed from the letters of the word "THINK" = 5*4*3*2*1=120

    --------------------------------------------------------------------------------------------------------------------------------

    Let us now try to calculate the no of words (that can be formed from the letters of the word "THINK") in which T and H always appear together.

    Consider TH as one entity. Then there are 4*3*2*1=24 ways.
    However, TH itself can be permuted in 2*1=2 ways.

    Thus there are 24*2=48 words in which T and H always appear together.

    --------------------------------------------------------------------------------------------------------------------------------

    So, the number of ways in which the word "THINK" can be arranged so that the "T" and the "H" are separated by at least one letter is 120-48=72.
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  3. #3
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    Thanks! Had to look up what permutation was but I'm pretty sure I get it now.

    Can someone please help me with the second question I'm having trouble with?
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