Results 1 to 9 of 9

Math Help - How many combinations of sequences can I make?

  1. #1
    Newbie
    Joined
    Jul 2012
    From
    Texas
    Posts
    7

    How many combinations of sequences can I make?

    Let's say I have 6 colored pencils: Pink, Purple, Yellow, Red, Blue, and Green.

    I want to fill a 25-line piece of paper with text. Each line has to be one color only.

    Each new line has to be a different color than the previous line. That is the only stipulation.

    How many pieces of paper could I produce with a unique sequence of colors?

    EDIT: Of course, this could be visualized many different ways, besides text on paper. In a nutshell, I'm looking for how many sequences of 25 items from a pool of 6 different items, where the next item in the list is different from the previous one.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,639
    Thanks
    1592
    Awards
    1

    Re: How many combinations of sequences can I make?

    Quote Originally Posted by jonarmani View Post
    EDIT: Of course, this could be visualized many different ways, besides text on paper. In a nutshell, I'm looking for how many sequences of 25 items from a pool of 6 different items, where the next item in the list is different from the previous one.
    How many choices do we have for the first item is this list of twenty-five?

    How many choices do we have for the second item is this list of twenty-five?

    So what is the final answer?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Joined
    Dec 2007
    From
    Ottawa, Canada
    Posts
    3,102
    Thanks
    68

    Re: How many combinations of sequences can I make?

    Can you tell us how many if 3 pencils and 7 lines only?
    Last edited by Wilmer; July 17th 2012 at 04:39 PM.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    Jul 2012
    From
    Texas
    Posts
    7

    Re: How many combinations of sequences can I make?

    The first one can be any of the 6. You don't have 25 choices, you have 25 slots to fill. See my colored pencil example.

    And I don't know the final answer... I need to know, which is why I'm posting this puzzle to a math help forum :-)

    Quote Originally Posted by Plato View Post
    How many choices do we have for the first item is this list of twenty-five?

    How many choices do we have for the second item is this list of twenty-five?

    So what is the final answer?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Jul 2012
    From
    Texas
    Posts
    7

    Re: How many combinations of sequences can I make?

    Quote Originally Posted by Wilmer View Post
    Can you tell us how many if 3 pencils and 7 lines only?
    Let's see...
    Red, green, blue, red, green, blue, red
    Red, green, blue, red, green, blue, green
    Red, green, blue, red, green, red, blue
    Red, green, blue, red, green, red, green
    Red, green, blue, red, blue, red, blue
    Red, green, blue, red, blue, red, green
    ....
    On and on... see what I mean? See the pattern?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,639
    Thanks
    1592
    Awards
    1

    Re: How many combinations of sequences can I make?

    Quote Originally Posted by jonarmani View Post
    See the pattern?
    Well yes, I saw what you meant the first time around,
    I hoped you could finish it for yourself.

    Quote Originally Posted by jonarmani View Post
    The first one can be any of the 6. You don't have 25 choices, you have 25 slots to fill. See my colored pencil example. And I don't know the final answer.
    The final answer: 6\cdot 5^{24}. WHY?
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Newbie
    Joined
    Jul 2012
    From
    Texas
    Posts
    7

    Re: How many combinations of sequences can I make?

    Ok Plato, I see how you got the 6 at least. In the attached image, I show a simplified version of the problem: A pool of 3 items in 5 slots. My drawing is all possibilities starting with item 1. The total number of possibilities would include a similar tree for sequences starting with 2, and then 3. The total number of sequences [paths through the tree] is equal to the number of nodes on the bottom row, which is 16 in this tree. Therefore, the total answer to this one is 3 * 16 = 48. Since my original puzzle involves 6 items, there will be 6 of these trees. That easily explains the 6 in your answer.

    Attachment 24310

    Now check out when I add more items:

    Attachment 24311

    It appears that at each depth, we get the number of items to the depth-th power...
    Row 1 = (items-1) ^ (Row-1)
    Row 2 = (items-1) ^ (Row-1)
    Row 3 = (items-1) ^ (Row-1)

    In my first tree, with 3 items:
    Row 1 = (3-1) ^ (1-1) = 2 ^ 0 = 1 node
    Row 2 = (3-1) ^ (2-1) = 2 ^ 1 = 2 nodes
    Row 3 = (3-1) ^ (3-1) = 2 ^ 2 = 4 nodes
    Row 4 --> 8 nodes
    Row 5 --> 16 nodes

    And this works out for that second tree as well:
    Row 1 = (4-1) ^ (1-1) = 3 ^ 0 = 1 node
    Row 2 = (4-1) ^ (2-1) = 3 ^ 1 = 3 nodes
    Row 3 = (4-1) ^ (3-2) = 3 ^ 2 = 9 nodes

    Ok! So this seems to work out! Amazing how drawing + charting can be so helpful!

    My answer will rely on the equation:
    N = (items-1) ^ (slots-1)

    So since I wanted to find the total number of sequences you could get from 6 items in 25 slots:
    N = (6-1) ^ (25-1) = 5 ^ 24

    And since there are six trees, the answer will be 6 times this, or 6 x 5^24 -- which is the answer you provided!

    Thanks for helping me understand this! Unfortunately, I can't write a program to quickly brute-force 358 quadrillion possibilities, but at least I know now to search for a better algorithm for the problem I'm having (and needed this solution for).
    Last edited by jonarmani; July 18th 2012 at 08:12 AM.
    Follow Math Help Forum on Facebook and Google+

  8. #8
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,639
    Thanks
    1592
    Awards
    1

    Re: How many combinations of sequences can I make?

    Quote Originally Posted by jonarmani View Post
    Ok Plato, I see how you got the 6 at least. In the attached image, I show a simplified version of the problem: A pool of 3 items in 5 slots. My drawing is all possibilities starting with item 1. The total number of possibilities would include a similar tree for sequences starting with 2, and then 3. The total number of sequences [paths through the tree] is equal to the number of nodes on the bottom row, which is 16 in this tree. Therefore, the total answer to this one is 3 * 16 = 48. Since my original puzzle involves 6 items, there will be 6 of these trees. That easily explains the 6 in your answer.
    .
    You wasted your time doing that reply. I said that I understood you edited OP.
    Quote Originally Posted by jonarmani View Post
    EDIT: Of course, this could be visualized many different ways, besides text on paper. In a nutshell, I'm looking for how many sequences of 25 items from a pool of 6 different items, where the next item in the list is different from the previous one.
    Lets generalize you question.
    How many sequences of N items from a pool of K different items, where the next item in the list is different from the previous one?
    The answer is K\cdot\(K-1)^{N-1} .

    Look at new example. A pool of 3 items in 5 slots. Here k=3~\&~N=5
    So 3\cdot 4^2=48.
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Newbie
    Joined
    Jul 2012
    From
    Texas
    Posts
    7

    Re: How many combinations of sequences can I make?

    Quote Originally Posted by Plato View Post
    You wasted your time doing that reply. I said that I understood you edited OP.
    I'm not sure what you're talking about. I didn't waste my time doing that, I wrote that out so I could have an easier example to illustrate with my drawings. You realize I didn't know the answer going into this, right? That last post of mine was me figuring the answer out.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Convergence in sequences of sequences
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: October 19th 2010, 07:28 AM
  2. Sequences and the sequences' arithmetics
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: October 6th 2010, 09:31 PM
  3. Monotone sequences and Cauchy sequences
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: March 21st 2009, 08:59 PM
  4. use only four 4's to make 10
    Posted in the Math Topics Forum
    Replies: 29
    Last Post: September 5th 2008, 10:08 AM
  5. Replies: 5
    Last Post: January 16th 2008, 04:51 PM

Search Tags


/mathhelpforum @mathhelpforum