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Permuations and Combinations Problem

  1. Nov 30, 2014 #1
    Can someone please explain the following question? I am also posting the solution but I didn't understand it well enough. Why is the answer (C)? Can someone explain it to me?


    In a shooting competition, three targets are set as shown :

    Condition :
    Target (I) has four rings on which a person can hit in order from inside to outside.
    Target (II) has three rings on which a person can hit in order from outside to inside.
    Target (III) has five rings on which a person can hit in order from inside to outside.

    The number of ways in which 12 shots (one at each ring) can be made :
    [Hint : Any target can be chosen before not completing specific target but order of hit for a particular target should be as specified above in condition.]

    (A) 12! /(4! 3! 5!)
    (B) 12!/(4! 3! 5!) x 3!
    (C) C(12,4) x C(8,3) x C(5,5)
    (D) 12!


    72.jpg 1.jpg
     
  2. jcsd
  3. Dec 1, 2014 #2

    mfb

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    Forget the rings for now - they get hit in the order of shots, who cares.

    The shot sequence can be written like "ABCCBCBACAC" where A, B, C are the three disks. How many of those sequences are there? You have C(12,4) ways to pick positions for the "B"s, ...
     
  4. Dec 2, 2014 #3
    Hey mfb, I didn't understand what you meant by the above statements. Could you please explain it more clearly?
     
  5. Dec 2, 2014 #4

    mfb

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    Which part is unclear?
     
  6. Dec 2, 2014 #5
    Why do we have C(12,4) ways to pick position for Bs?

    Okay, I think I got this now. Suppose one of the possible sequence of shots is-
    AAAABBBCCCCC (Suppose we complete the disk fully which we started). Now the number of ways to arrange this word is 12!/(4!x3!x5!) or C(12,4) (select 4 places from the 12 above) x C(8,3) (select 3 places from the 8 left above now) x C(5,5) (select 5 places from the 5 left now)

    Is this the correct explanation?
     
  7. Dec 2, 2014 #6

    mfb

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    You have 12 possible positions, and you have to choose 4.

    Right.
     
  8. Dec 2, 2014 #7
    Okay! Thanks alot for the help!
     
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