Forum — Daily Challenge
    • Categories
    • Recent
    • Tags
    • Popular
    • Users
    • Groups
    • Login

    Combinatorics Problem

    Math Problems
    2
    2
    416
    Loading More Posts
    • Oldest to Newest
    • Newest to Oldest
    • Most Votes
    Reply
    • Reply as topic
    Log in to reply
    This topic has been deleted. Only users with topic management privileges can see it.
    • G
      generousseagull M0 M1 M2 M3 M4
      last edited by

      How do I solve this:

      "Pat is to select six cookies from a tray containing only chocolate chip, oatmeal, and peanut butter cookies. There are at least six of each of these three kinds of cookies on the tray. How many different assortments of six cookies can be selected?"

      The answer is shown below, but I don't know how to get the answer. Please explain. Your help is much appreciated.

      answer: 28

      1 Reply Last reply Reply Quote 0
      • J
        JoyfulSapling M2★
        last edited by JoyfulSapling

        Explanation:
        Of the 6 cookies Pat chooses, there are three types: chocolate chip, oatmeal, and peanut butter cookies. We also note that there can be 0 of a type. We can use the stars and bars formula to divide 6 objects into three categories (n is the number of objects, which is cookies in this case, and k is the number of categories):

        n+k-1 choose k-1
        = 6+3-1 choose 3-1
        = 8 choose 2
        = 28

        That is why the answer is 28.

        Please find below the proof for the stars and bars formula.
        Let's use the cookies problem from above to help us with the proof. Right now, n=6 (6 cookies) and k=3 (3 types of cookies).
        Putting 6 cookies into 3 categories looks like this:
        costume1.png
        There are 8 objects total in that picture (6 cookies + 2 bars).
        Let's say we remove all objects and hold them. There are 8 spots to put our 8 objects. We can choose 2 spots of these 8 spots to put our bars, and the cookies go into the remaining 6 spots. Therefore, there are 8 choose 2 = 28 different combinations of 6 cookies.

        So, in the formula n+k-1 choose k-1, n+k-1 represents the total number of spots to put the objects and the bars, and k-1 represents the number of bars needed to split the objects into k categories (for example, in our problem, 2 bars were needed to split the 6 cookies into 3 categories).

        Power of a Person

        1 Reply Last reply Reply Quote 0

        • 1 / 1
        • First post
          Last post
        Daily Challenge | Terms | COPPA