How To Construct And Properties Of Pascal's Triangle

Introduction

One of the most interesting number pattern in mathematics is Pascal's triangle. It was developed by a famous french mathematician and philosopher, Blaise Pascal.

Construct Pascal's triangle

To construct Pascal's triangle start with 1 at top, then continue placing numbers below it in a triangular pattern. The two side edges always have 1s placed on them. Each number in triangle is the sum of two numbers just around it in the above row.

Every number in the bottom row is the sum of the two numbers which are placed around it in the row above it.

To start with, 1s are placed on the first and second rows as shown in the following figure.

Numbers on Pascal's triangle

2 in the third row is the sum of 1 and 1 placed around it in the second row.

3 in the fourth row is the sum of 1 and 2 that are placed in third row.

4 in the fifth row is the sum 1 and 3 placed around it in the fourth row.
Similarly, 6 in the fifth row is the sum 3 and 3 placed in the fourth row.

15 in the seventh row is the sum of 5 and 10 placed in the sixth row.
Also, 20 in the 7^th row is sum of numbers 10 and 10 that are placed in the sixth row.

7 in the 8^th row is sum 1 and 6 around it in the 7^th row.
21 is sum of 6 and 15 and 35 is sum of 20 and 15 placed in 7^th row.

Number pattern on the diagonals

Pascal's triangle diagonals

Number pattern formed by numbers placed diagonally in Pascals triangle

The above figure, shows an interesting pattern formed by the numbers those are placed diagonally.

The pattern of numbers on diagonals of pascal's triangle
Diagonal	Numbers on diagonal	Type of numbers
1^st	1, 1, 1, 1, 1, 1, 1, 1, 1	only 1s
2^nd	1, 2, 3, 4, 5, 6, 7, 8	natural
3^rd	1, 3, 6, 10, 15, 21,	triangular
4^th	1, 4, 10, 20, 35,	tetrahedral
5^th	1, 5, 15, 35,	pentatope
6^th	1, 6, 21,	hexatope

It can be noted from the above table that the numbers on every diagonal of pascal's triangle makes special patterns which match to natural, triangular, tetrahedral, pentatope and hexatope numbers.

Properties of sums of numbers on rows

The sum of numbers on the rows of Pascal's triangle makes up the power of 2, which is the property of sums of numbers on rows.
Let's understand it from the following figure.

Property of row sums

Property of sums of numbers placed on rows in Pascals triangle

The following table shows how the sum of numbers on a row of pascal's triangle can be written into power of 2.

Sum of numbers on a row of pascal's triangle
Row	Numbers	Sum	Sum equals to	Power of 2
1^st	1	1	1	$2^{0}$
2^nd	1, 1	1 + 1	2	$2^{1}$
3^rd	1, 2, 1	1 + 2 + 1	4	$2^{2}$
4^th	1, 3, 3, 1	1 + 3 + 3 + 1	8	$2^{3}$
5^th	1, 4, 6, 4, 1	1 + 4 + 6 + 4 + 1	16	$2^{4}$
6^th	1, 5, 10, 10, 5, 1	1 + 5 + 10 + 10 + 5 + 1	32	$2^{5}$
7^th	1, 6, 15, 20, 15, 6, 1	1 + 6 + 15 + 20 + 15 + 6 + 1	64	$2^{6}$
8^th	1, 7, 21, 35, 35, 21, 7, 1	1 + 7 + 21 + 35 + 35 + 21 + 7 + 1	128	$2^{7}$