Multinomial Theorem: Find Coefficient of x^{12}y^{24}

  • Thread starter blinktx411
  • Start date
  • Tags
    Theorem
In summary, using the multinomial theorem, the coefficient of the x^{12}y^{24} term for (x^3+2xy^2+y+3)^{18} can be found by setting x_1=x^3, x_2=2xy^2, x_3=y, x_4=3 and finding the values of k_1, k_2, k_3, and k_4 that satisfy the equations 3k_1+k_2 = 12, 2k_2+k_3 = 24, k_1+k_2+k_3+k_4=18, and k_i >= 0. This results in k_1 being either
  • #1
blinktx411
35
0

Homework Statement


Find the coefficient of the [tex] x^{12}y^{24} [/tex] for [tex] (x^3+2xy^2+y+3)^{18} [/tex].


Homework Equations


Multinomial theorem, as stated on http://en.wikipedia.org/wiki/Multinomial_theorem


The Attempt at a Solution


Using the multinomial theorem in the form of the wikipedia post, I would set [tex] x_1=x^3, x_2=2xy^2, x_3=y, x_4=3 [/tex]. Now I need to find the k's that "match" the coefficients of the term from the problem statement. This will give me a relation between the k's of the following form
[tex] 3k_1+k_2 = 12, 2k_2+k_3 = 24, and k_1+k_2+k_3+k_4=18 [/tex]. Now I let [tex] k_1 [/tex] vary and record the discrete values [tex] k_2, k_3, k_4 [/tex], but what I'm confused on is why do I sometimes get negative values for [tex] k_4 [/tex]? Do I need to do something different with the relationship of the k's. Thanks in advance.
 
Physics news on Phys.org
  • #2
You wrote down three restrictions, but there is a fourth as well, namely that all the k_i's must be nonnegative. Looks like k_1 can be 0 or 1, and that's all.
 
  • #3
cool, so everything else looks good?
 

1. What is the Multinomial Theorem?

The Multinomial Theorem is a mathematical formula used to expand the power of a multinomial expression. It is similar to the Binomial Theorem, but can be applied to expressions with more than two terms.

2. How do you find the coefficient of a specific term in a multinomial expression using the Multinomial Theorem?

In order to find the coefficient of a specific term, you must use the formula (n1+n2+...+nk)! / (n1!n2!...nk!), where n1, n2, etc. represent the powers of each term in the expression and k represents the total number of terms in the expression. In our example of finding the coefficient of x^{12}y^{24}, we would have to find the coefficient using the formula (12+24)! / (12!24!).

3. What is the significance of the exponent in the Multinomial Theorem?

The exponent in the Multinomial Theorem represents the number of terms in the expression. It is used in the formula to determine the number of possible combinations for each term.

4. Can the Multinomial Theorem be used to expand expressions with variables other than x and y?

Yes, the Multinomial Theorem can be used to expand expressions with any number of variables. The formula remains the same, but the variables and their corresponding powers must be substituted into the formula.

5. What is the practical application of the Multinomial Theorem?

The Multinomial Theorem is primarily used in mathematics and statistics, particularly in the fields of probability and combinatorics. It is also used in various other areas of science and engineering, such as in the analysis of data, in computer science, and in physics.

Similar threads

  • Calculus and Beyond Homework Help
Replies
1
Views
2K
Replies
13
Views
1K
  • Calculus and Beyond Homework Help
Replies
6
Views
761
  • Calculus and Beyond Homework Help
Replies
1
Views
904
  • Calculus and Beyond Homework Help
Replies
3
Views
1K
Replies
3
Views
618
  • Advanced Physics Homework Help
Replies
2
Views
1K
  • Introductory Physics Homework Help
Replies
5
Views
728
  • Differential Equations
Replies
1
Views
1K
  • Advanced Physics Homework Help
Replies
3
Views
2K
Back
Top