Solved

binomial theorem

Posted on 2013-11-30
11
232 Views
Last Modified: 2013-12-01
Is the number of terms in any binomial expansion always one more than the degree?

So, (x + y) ^ 10 - number of terms is 11.   And also 11 for  (x + y +2z) ^ 10 ??

And for (w + 2x + y -5z) ^ 12 - number of terms is 13.

Also, I know how to find the coefficient of any term where there are only 2 factors, like:

(2s -t) ^ 12

But how do I find the coefficient of the x^2 y^3 z^5 term for (x + y + z) ^ 10 ?
0
Comment
Question by:HLRosenberger
  • 8
  • 3
11 Comments
 
LVL 27

Expert Comment

by:d-glitch
ID: 39687052
Is the number of terms in any binomial expansion always one more than the degree?
YES

So, (x + y) ^ 10 - number of terms is 11.
YES

And also 11 for  (x + y +2z) ^ 10 ??
NO.  This has three terms so it is not a binomial expansion.
0
 
LVL 27

Expert Comment

by:d-glitch
ID: 39687056
http://en.wikipedia.org/wiki/Trinomial_expansion


But how do I find the coefficient of the x^2 y^3 z^5 term for (x + y + z) ^ 10 ?
10!/(2! 3! 5!)  =  2520
0
 
LVL 27

Expert Comment

by:d-glitch
ID: 39687147
From the Pythagorean Pyramid:

 (x + y +2z) ^ 10    will have 10*(10+1)/2  =  55 terms

 (w + 2x + y -5z) ^ 12 -  will have 12*(12+1)/2  =  78 terms
0
Gigs: Get Your Project Delivered by an Expert

Select from freelancers specializing in everything from database administration to programming, who have proven themselves as experts in their field. Hire the best, collaborate easily, pay securely and get projects done right.

 
LVL 1

Author Comment

by:HLRosenberger
ID: 39687156
d-glitchP

How do I find the number of term using a formula?
0
 
LVL 27

Expert Comment

by:d-glitch
ID: 39687183
The formula for trinomial coefficients is in the Wikipedia article.

I used it to find one of the 55 terms in the expansion of    (x + y + z) ^ 10

        2520 * x² * y³ * z^5

I can find the coefficient for any other term you specify the same way.

What exactly are you trying to do?
0
 
LVL 27

Accepted Solution

by:
d-glitch earned 500 total points
ID: 39687288
Think about the Pythagorean Pyramid and the expansion of  (x + y + z) ^ n

The top (0th) level of the Pyramid has one term:         (x + y + z) ^ 0  =  1

The next (1st) level of the Pyramid has three terms:   (x + y + z) ^ 1  =  x + y + z

The next (2nd) level of the Pyramid has six terms:      
            (x + y + z) ^ 2  =  x² + 2xy + 2xz + y² + 2yz + z²
==========================================================================

Picture each level of the Pyramid as a triangle with x at the top, y at the bottom left, and z at the bottom right.

The 10th level of the Pyramid has 55 terms.  The triangle that makes up this level has 11 rows.

The frist row has one term:          x^10

The second row has two terms:   x^9 y  +  x^9 z          

[Starting here I am ignoring coefficients.  Use the formula.]

The third row has three terms:    x^8 y²  +  x^8 y z  +  x^8 z²
.  .  .  .

The tenth row has ten terms:       x y^9  +  x y^8 z  +  . . .  +  x y z^8  +  x z^9

The last row has eleven terms:     y^10  +   y^9 z    +  . . .  +   y z^9   +    z^10
0
 
LVL 1

Author Closing Comment

by:HLRosenberger
ID: 39687692
Thanks so Much!
0
 
LVL 1

Author Comment

by:HLRosenberger
ID: 39687698
One more question for :

 (w + 2x + y -5z) ^ 12 -  will have 12*(12+1)/2  =  78 terms

How did you come up with 12*(12+1)/2?  What's the general formula the number of terms for a multinomial expansion?
0
 
LVL 27

Expert Comment

by:d-glitch
ID: 39688246
(w + 2x + y -5z) ^ 12 -  will have 12*(12+1)/2  =  78 terms

No.    It will have.   78 +66 +55 +45 +36 +21 +15 +10 +6 +3 +1  terms.

(x)^n   will have 1  term

(x + y)^n   will have n+1  terms

(x + y +z)^n   will have (n+1)(n+2)/2  terms.
This is the sum of the integers from 1 to n+1.  These are triangonal numbers.

(w + x + y +z)^n   will have on the order of n^3 terms.
It will be the sum of the first n+1 triangonal numbers.  These are pyramidal numbers.

Each higher level will build on the one below.
0
 
LVL 27

Expert Comment

by:d-glitch
ID: 39688251
Correction

No.    It will have.   78 +66 +55 +45 +36 +28 +21 +15 +10 +6 +3 +1  terms.
0
 
LVL 27

Expert Comment

by:d-glitch
ID: 39688368
More corrections:

(x + y +2z) ^ 10    will have  (n+1)(n+2)   =   11*(11+1)/2  =  66 terms

 (w + 2x + y -5z) ^ 12 -  will have 13*(13+1)/2  =  91 terms
0

Featured Post

Live: Real-Time Solutions, Start Here

Receive instant 1:1 support from technology experts, using our real-time conversation and whiteboard interface. Your first 5 minutes are always free.

Question has a verified solution.

If you are experiencing a similar issue, please ask a related question

Complex Numbers are funny things.  Many people have a basic understanding of them, some a more advanced.  The confusion usually arises when that pesky i (or j for Electrical Engineers) appears and understanding the meaning of a square root of a nega…
This article provides a brief introduction to tissue engineering, the process by which organs can be grown artificially. It covers the problems with organ transplants, the tissue engineering process, and the current successes and problems of the tec…
This is a video describing the growing solar energy use in Utah. This is a topic that greatly interests me and so I decided to produce a video about it.
Finds all prime numbers in a range requested and places them in a public primes() array. I've demostrated a template size of 30 (2 * 3 * 5) but larger templates can be built such 210  (2 * 3 * 5 * 7) or 2310  (2 * 3 * 5 * 7 * 11). The larger templa…

786 members asked questions and received personalized solutions in the past 7 days.

Join the community of 500,000 technology professionals and ask your questions.

Join & Ask a Question