1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
One of the uses of this triangle is to expand algebraic expressions called binomials.
For example:
(x+y)^4 = x^4 + 4x^3 * y + 6x^2 * y^2 + 4x * y^3 + y^4
The numbers in front of the x and y terms in the binomial expansion above are the coefficients in the fourth row of Pascal's triangle (assuming you consider the top row to be row zero).
You could also multiply and distribute this expression (x+y)(x+y)(x+y)(x+y), however this method is longer,tedious, and unnecessary if you already know about Pascal's triangle and the binomial theorem.
The coefficients in Pascal's Triangle are known as the binomial coefficients. Binomial coefficients can be written as n C r where n is the number of items in a set and r is the number of items you are "choosing" from the set. Using binomial coefficient notation, we can write the first 8 rows of Pascal's triangle as
0 C 0
1 C 0 1 C 1
2 C 0 2 C 1 2 C 2
3 C 0 3 C 1 3 C 2 3 C 3
4 C 0 4 C 1 4 C 2 4 C 3 4 C 4
5 C 0 5 C 1 5 C 2 5 C 3 5 C 4 5 C 5
6 C 0 6 C 1 6 C 2 6 C 3 6 C 4 6 C 5 6 C 6
7 C 0 7 C 1 7 C 2 7 C 3 7 C 4 7 C 5 7 C 6 7 C 7
If you use the formula n C r = n! / (n-r)! r! with all of the binomial coefficients in the above triangle, your result will be the original triangle at the top of this post.
Expanding algebraic expressions is not the only use for binomial coefficients. Binomial coefficients are also useful and ubiquitous in probability and statistics.