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Difference between at least and exactly? combinatorics!

  1. Nov 12, 2006 #1
    Hello everyone.

    I'm revewing these 2 problems and im' not seeing how they are getting the answer for the "...exactly 4 penny's"

    Here's the 2 problems:

    (a) A large pile of coins consist of quarters, dimes, nickels, and pennies (at least 20 of each). How many different collections of 20 coins can be chosen if AT LEAST 4 pennies must be chosen?

    Well you have 4 categories, Quaters, Dimes, nickles and Pennies it says (at least 20 of each).

    I'm going to represent the categories as bars |'s and the coins as x's.
    If the problem said, if at least 4 pennies are already chosen you can assume you already put 4 x's in the penny category, now you have 16 places left to put coins, anywhere you want.

    so you could have like this:
    xxxxx|xxxxx|xxxx|xx
    -------------------
    xxxx
    P N Q D
    The above ---- means the coins u distrubted, and the stuff under the --- means the coins already there.
    So if you have 4 categories thats represented by n-1 bars (3) and 16 x's. So you have a total of

    (16+3 choose 16) = 969 which is the correct answer.


    Now for the second question:
    (b) How many different collections of 20 coins can be chosen if EXACTLY 4 pennies must be chosen?

    THe work shown is, they only used 2 bars |'s and 16 x's and got:

    (16 + 2 choose 16) = (18 choose 16) = 153.

    Now why did they only have 3 categories now isntead of 4? The reason I say they only had 3 is because they had 2 bars, thus they had to have 2+1 categories.

    Why in the first problem did they have to use 4 categories, is it because they said "at least 20 of each?"

    and in the 2nd problem they only said, If you have exactly 4 penny's, how many different collections of 20 coins can be chosen?

    I thought to myself, well if only 4 penny's are allowed the minimum amount of categories to use and still put 20 coins in would be to have a category of Penny's which is going to have 4 x's, now your going to have 16 x's (coins) to put in any category you want. So why couldn't u put 4 x's in penny's and 16 in dimes?

    Then you would only have 2 categories, so 2-1 = 1 bar. Then you would have

    (16 + 1 choose 16) = (17 choose 16)

    But the answer is (16+2 choose 16) = (18 choose 16), why did they have to use 3 categories?


    THanks!
     
  2. jcsd
  3. Nov 12, 2006 #2
    The reason is this. In the first problem you choose 4 pennies, then you have 16 coins left to pick, and these coins can be any of the 4 choices (quarters, dimes, nickels, pennies). In the second problem you choose 4 pennies, then you have 16 coins left to pick, but none of these remaining 16 coins can be pennies, thus you are picking from 3 (quarters, dimes, nickels).
     
  4. Nov 12, 2006 #3
    Oooo hah i got it!
    thanks again matt!
     
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