Results 1 to 4 of 4

Math Help - SAT Math Counting Problem

  1. #1
    Newbie
    Joined
    Sep 2010
    Posts
    3

    SAT Math Counting Problem

    John went to dinner and had a bill of 2010 dollars. He only has 1,2,5, and 10 dollar bills. How many different ways can he pay for dinner.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor undefined's Avatar
    Joined
    Mar 2010
    From
    Chicago
    Posts
    2,340
    Awards
    1
    Quote Originally Posted by altamash View Post
    John went to dinner and had a bill of 2010 dollars. He only has 1,2,5, and 10 dollar bills. How many different ways can he pay for dinner.
    Here is one way.

    How many ways can we spend 2, 5, and 10 dollar bills using at most 2010 dollars? The 1 dollar bills are used to make up any difference. Include 0 as valid way to spend 2,5,10.

    Fix the number of 10 dollar bills as 0. Then we spend at most 2010 dollars with 2 and 5. Fix # of 5s at 0, then # ways is 2010/2+1. Fix # of 5's at 1 then # ways is floor(2005/2)+1. Write the sum.

    Now fix number of 10 dollar bills as 1. We spend at most 2000 with 2 and 5. Fix # 5 at 0, get 2000/2+1. Etc. Write sum.

    Compute sum of sums.
    Last edited by undefined; September 4th 2010 at 09:26 AM. Reason: Forgot the "+1"
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,802
    Thanks
    692
    Hello, altamash!

    I don't have a solution, just another approach.


    John went to dinner and had a bill of 2010 dollars.
    He only has one-, two-, five-, and 10-dollar bills.
    How many different ways can he pay for dinner?


    \text{Let: }\;\begin{Bmatrix} T &=& \text{number of tens} \\ F &=& \text{number of fives} \\ D &=& \text{number of "deuces"} \\ S &=& \text{number of singles} \end{Bmatrix}


    We have the equaton: . 10T + 5F + 2D + S \:=\:2010

    . . and we want the number of nonnegative integral solutions.



    I'll wait in the car . . .



    By the way, what was the main course of John's dinner?
    . . White truffles drenched in saffron sauce?
    (Or maybe he went to Wendy's and is a big tipper?)
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    Sep 2010
    Posts
    3
    lol soraban, i was wondering what he had for dinner also. The thing is the problem was originally just 2,5, and 10 dollar bills. That was easy because there were no combination of adding 2 and 5 together to make a mulitple of ten. so the solution was 20503. but with the introduction of 1 dollar bills you can add to the two dollar bills and the 5 dollar bills. This will lead to a huger combination, and i don't see a pattern.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. The Math Behind Card Counting
    Posted in the Statistics Forum
    Replies: 2
    Last Post: May 8th 2011, 07:48 PM
  2. Counting problem
    Posted in the Discrete Math Forum
    Replies: 3
    Last Post: December 23rd 2010, 03:24 AM
  3. Counting, Probability, Trees, etc in Discrete Math
    Posted in the Discrete Math Forum
    Replies: 16
    Last Post: December 1st 2010, 12:58 PM
  4. Replies: 3
    Last Post: April 13th 2009, 05:42 PM
  5. Counting - Discrete math
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: November 17th 2008, 10:31 AM

Search Tags


/mathhelpforum @mathhelpforum