How does Pascal triangle apply to (a+b+c)^n and (a+b+c+...+d)^n?

In summary, Pascal's triangle is not used in the equation (a+b+c)^n, but rather in (a+b)^n. To solve for "m" numbers to the power "n," you would need to use multinomial coefficients rather than binomial coefficients. The formula for multinomial coefficients is (i+j+...+k)!/(i! j!...k!), where "i," "j," and "k" represent different numbers.
  • #1
MathematicalPhysicist
Gold Member
4,699
371
does pascal triangle use in this equation (a+b+c)^n i know it is used in (a+b)^n?

and how could you solve for m number of numbers to the power n?
(a+b+c+...+d)^n
||
\/
m numbers.
 
Mathematics news on Phys.org
  • #2
No, Pascal's triangle give binomial coefficients.

What you need are "multinomial" coefficients.

The binomial coefficients are given by nCm= n!/(m!(n-m)!) because there are that many ways of arranging m x's and n-m y's to give the product xmyn-m.

The "trinomial" coefficient for xiyjzk would be (i+j+k)!/(i! j! k!)

If you have "m" numbers to the "n" power: (x1+ x2+...+xm)n then the "multi-nomial" coefficient for x1ixjj...xmk would be

(i+ j+ ...+ k)!/(i! j! ... k!).
 
  • #3
thanks :smile:
 

What is a Pascal triangle?

A Pascal triangle is a geometric arrangement of numbers in the shape of a triangle. It is named after the French mathematician Blaise Pascal.

How is a Pascal triangle constructed?

A Pascal triangle is constructed by starting with a single "1" at the top, and then each subsequent row is created by adding the two numbers directly above it. For example, the third row would be 1+1=2, the fourth row would be 1+2=3, and so on.

What are the properties of a Pascal triangle?

Some key properties of a Pascal triangle include: each number is the sum of the two numbers above it, the sum of the numbers in each row is equal to 2^n (where n is the row number), and the triangle is symmetrical.

What is the significance of a Pascal triangle?

Pascal triangles have numerous applications in mathematics, including in probability, binomial expansion, and the study of patterns and sequences. They can also be used to solve various mathematical problems and are often used in computer programming.

Are there any variations of a Pascal triangle?

Yes, there are many variations of a Pascal triangle, including the Sierpinski triangle, the Yang Hui triangle, and the Tartaglia triangle. These variations have different patterns and properties, but are all based on the same concept of adding the numbers above to create a new row.

Similar threads

Replies
0
Views
66
Replies
1
Views
1K
Replies
1
Views
376
  • General Math
Replies
1
Views
752
  • General Math
Replies
1
Views
992
Replies
1
Views
758
  • General Math
Replies
2
Views
1K
Replies
35
Views
2K
Replies
1
Views
757
Back
Top