A community for students.

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

ParthKohli

  • one year ago

Hey.

  • This Question is Closed
  1. ParthKohli
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Why is the number of solutions to the following equation:\[x_1 + x_2 + \cdots + x_k=n\]given \(a_i <x_i < b_i \) equal to the coefficient of \(x^{n}\) in the expansion of\[\prod_{i=1}^{n} \sum_{k=a_i}^{b_i}x^{k}\]

  2. imqwerty
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    number of solutions =\[x_{i} \]?

  3. ParthKohli
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Number of solutions = ordered pairs.

  4. ParthKohli
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    @ganeshie8

  5. ParthKohli
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    yeah

  6. ParthKohli
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    ah, kinda makes sense to me now.

  7. imqwerty
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    ahhgh that given expression is so bad..

  8. ParthKohli
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    I understood why. Now as an example if we want to calculate solutions of\[x_1 + x_2 +x_3=20\]subject to \(x_i \ge -3\) then\[(x^{-3} + x^{-2} + \cdots + x^{26})^3\]coefficient of \(x^{20}\)

  9. ParthKohli
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    \[\left(x^{-3}\cdot \left(\frac{1 - x^{30}}{1 - x}\right)\right)^{3}\]

  10. ParthKohli
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    ^ how do we calculate the coefficient of \(x^{20}\) in that?

  11. imqwerty
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    i didn't get that 2nd step? (x^-3 +x^-2..x^26)^3

  12. ParthKohli
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    ah, the lower restriction is given to us: \(x_1, x_2, x_3 \ge -3\) since their sum is \(20\), the max. value any of them can take is \(26\) (if and only if the other two are -3) thus \(-3 \le x_i \le 26\)

  13. ParthKohli
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    now refer to the above identity

  14. imqwerty
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    :o ok now i get the ques :)

  15. ParthKohli
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    but how do we calculate the coefficient? :(

  16. ParthKohli
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    ah, since \(1 -x^{30}\) cannot generate \(x^{20}\) we can remove that and find the coefficient of \(x^{20}\) in \(x^{-9} \cdot \dfrac{1}{(1-x)^3}\)

  17. ParthKohli
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    I'm not sure about the above step... can you confirm it?

  18. imqwerty
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    wait what was that equation where did it go? http://prntscr.com/8nkh3q

  19. ParthKohli
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    maybe there are displaying problems... refresh?

  20. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    generating functions

  21. Not the answer you are looking for?
    Search for more explanations.

    • Attachments:

Ask your own question

Sign Up
Find more explanations on OpenStudy
Privacy Policy

Your question is ready. Sign up for free to start getting answers.

spraguer (Moderator)
5 → View Detailed Profile

is replying to Can someone tell me what button the professor is hitting...

23

  • Teamwork 19 Teammate
  • Problem Solving 19 Hero
  • You have blocked this person.
  • ✔ You're a fan Checking fan status...

Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.

This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.