Pascal's Triangle

Pascal's triangle is a triangular array constructed by summing adjacent elements in preceding rows. Pascal's triangle contains the values of the binomial coefficient. It is named after the $17^\text{th}$ century French mathematician, Blaise Pascal (1623 - 1662).

$$1\\ 1\quad 1\\ 1\quad 2 \quad 1\\ 1\quad 3 \quad 3 \quad 1\\ 1\quad 4 \quad 6 \quad 4 \quad 1\\ 1 \quad 5 \quad 10 \quad 10 \quad 5 \quad 1\\ \cdots$$

Pascal's triangle can be used to visualize many properties of the binomial coefficient and the binomial theorem.

Begin by placing a $1$ at the top center of a piece of paper. The next row down of the triangle is constructed by summing adjacent elements in the previous row. Because there is nothing next to the $1$ in the top row, the adjacent elements are considered to be $0:$

This process is repeated to produce each subsequent row:

This can be repeated indefinitely; Pascal's triangle has an infinite number of rows:

The topmost row in Pascal's triangle is considered to be the $0^\text{th}$ row. The next row down is the $1^\text{st}$ row, then the $2^\text{nd}$ row, and so on. The leftmost element in each row is considered to be the $0^\text{th}$ element in that row. Then, to the right of that element is the $1^\text{st}$ element in that row, then the $2^\text{nd}$ element in that row, and so on.

With this convention, each $i^\text{th}$ row in Pascal's triangle contains $i+1$ elements.

The convention of beginning the order with $0$ may seem strange, but this is done so that the elements in the array correspond to the values of the binomial coefficient.

Let $x_{i,j}$ be the $j^\text{th}$ element in the $i^\text{th}$ row of Pascal's triangle, with $0\le j\le i$. Then,

$$x_{i,j}=\binom{i}{j}.$$

This property of Pascal's triangle is a consequence of how it is constructed and the following identity:

Let $n$ and $k$ be integers such that $1\le k\le n$. Then,

$$\binom{n}{k}=\binom{n-1}{k-1}+\binom{n-1}{k}.$$

Using Pascal's triangle, what is $\binom{6}{2}$?

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Look for the $2^\text{nd}$ element in the $6^\text{th}$ row. The value of that element will be $\binom{6}{2}$.

Thus, $\binom{6}{2}=15$. $_\square$

Pascal's triangle is a way to visualize many patterns involving the binomial coefficient. Here are some of the ways this can be done:

Binomial Theorem

The $n^\text{th}$ row of Pascal's triangle contains the coefficients of the expanded polynomial $(x+y)^n$.

Expand $(x+y)^4$ using Pascal's triangle.

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The $4^\text{th}$ row will contain the coefficients of the expanded polynomial.

$$(x+y)^4=\color{#3D99F6}{1}x^4+\color{#3D99F6}{4}x^3y+\color{#3D99F6}{6}x^2y^2+\color{#3D99F6}{4}xy^3+\color{#3D99F6}{1}y^4$$

Hockey Stick Identity

Start at any of the "$1$" elements on the left or right side of Pascal's triangle. Sum elements diagonally in a straight line, and stop at any time. Then, the next element down diagonally in the opposite direction will equal that sum.

If you start at the $r^\text{th}$ row and end on the $n^\text{th}$ row, this sum is

$$\sum\limits_{k=r}^{n}\binom{k}{r}=\binom{n+1}{r+1}.$$

Using Pascal's triangle, what is $\displaystyle\sum\limits_{k=2}^{5}\binom{k}{2}?$

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Start at a $1$ on the $2^\text{nd}$ row, and sum elements diagonally in a straight line until the $5^\text{th}$ row:

$$1+3+6+10=20.$$

Or, simply look at the next element down diagonally in the opposite direction, which is $20$. $_\square$

Powers of 2

The sum of the elements in the $n^\text{th}$ row of Pascal's triangle is equal to $2^n$.

This is a way to express the identity

$$\sum\limits_{k=0}^{n}\binom{n}{k}=2^n.$$

The following property follows directly from the hockey stick identity above:

Triangular Numbers

The $2^\text{nd}$ element in the $(n+1)^\text{th}$ row is the $n^\text{th}$ triangular number.

This is a way to express the identity

$$\sum\limits_{k=1}^{n}{k}=\binom{n+1}{2}.$$

Sierpinski Gasket

Construct a Pascal's triangle, and shade in even elements and odd elements with different colors. The shading will be in the same pattern as the Sierpinski Gasket:

This is an application of Lucas's theorem.