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Combinatorial Problem of Letters

  1. Dec 14, 2011 #1
    1. The problem statement, all variables and given/known data

    From this combination of letters AAAXYZNO
    Find how many ways to pick 3 letters if the order does not matter.

    3. The attempt at a solution
    I tried to elaborate it like this:

    We have ___ 3 empty spaces.

    A__ (Two empty space for other different letters) -> 5C2 = 10
    AA_ (One empty space) -> 5C1 = 5
    AAA (All AAA) -> 1 way only.
    ___ (No A) -> 5C3 = 10

    The consideration is __A and A__ will be just the same because the order does not matter.

    Hence, total ways = 26.

    Or another way is simply 6C3 because we eliminate all of the As giving 20 ways only.

    I also want to ask if you guys know the insight that can be shared in solving this kind of problem since I feel the concept is just floating in my mind without concrete standing.
     
    Last edited: Dec 14, 2011
  2. jcsd
  3. Nov 12, 2016 #2

    haruspex

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    26 is right, and that is probably the best method.
    I did not understand the logic behind 6C3. You could use that to count all the ways with at most one A, I suppose,but you still need to add in the two and three A cases.
     
  4. Nov 12, 2016 #3

    lurflurf

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    The three way I see are
    5C0+5C1+5C2+5C3=1+5+10+10=26
    6C1+6C3=6+20=26
    8C3-6C2-6C2=8C3-6P2=56-30=26
    whichever you like best
     
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