# Pascal's Triangle
**A visual arrangement of the [[Binomial Theorem#Binomial coefficient|binomial coefficients]] in an infinite triangle.**
![[Pascal'sTriangle.svg|350]]
*The first 6 rows of Pascal's triangle*
*Pascal's triangle* visually represents the patterns of the binomial coefficients. The row and column indices are notated as $n$ and $k$ and start from zero.
It can be constructed by beginning with a $1$ at the top with each entry of subsequent rows being the *sum of the two entries immediately to the top-left and top-right*. All values of entries outside the triangle can be thought of as $0$.
![[Pascal'sTriangleConstruction.svg|300]]
## Formula
The entries of Pascal's triangle can also be filled with a [[recurrence relation]].
In the $n$th row, the $k$th entry is given by the [[Binomial Theorem#Binomial coefficient|binomial coefficient]] $\binom{n}{k}$. The coefficient can be determined with its formula for recursively from other entries of the triangle.
$\binom{n}{k}=\binom{n-1}{k-1}+\binom{n-1}{k}$
An alternate formula allows each successive entry of a row to be more easily calculated.
$\binom{n}{k}=\binom{n}{k-1}\times\frac{n+1-k}{k}$
## Patterns
### Coefficients of binomial expansions
The *coefficients* of a binomial expansion to the power of $n$ are the entries of the $n$th row of Pascal's triangle.
![[Pascal'sTriangleBinomialExpansion.svg]]
*The expansion of the power of four of a binomial and rows $2$ to $5$ of Pascal's triangle*
### Figurate numbers of simplexes
The *diagonals* in both directions of Pascal's triangle starting from the row $n = 2$ contain the [[simplex]] numbers of consecutive dimensions.
The first diagonals contain only $1