Not Just a Boring Triangle

Published by Ganit Charcha | Category - Math Articles | 2016-03-07 08:00:29

One Saturday afternoon Rima was relaxing with a storybook in her hand. Just then there was a knock on the door, her sister…oops…how could she forget?  She had to find a topic for her little sister’s Maths project in her school. Immediately she sat down in front of her laptop. After some moments of browsing she tried to reflect on what her tutor had told her two days back. She typed the words in Google search and started reading. A simple triangle formation of numbers had such an inside story! Who could guess? She became engrossed totally.

Pascal's Triangle: Pascal's Triangle, named after the famous French Mathematician Blaise Pascal, refers to one of the most interesting and absorbing number patterns.  We start with "$1$" at the top. And then each successive row will have one more element compared to its immediate previous row. This way of  placing numbers gives rise to a triangular pattern. 

Number to be placed at a particular position in a row is computed by adding the numbers directly above it in the previous row as has been shown in the diagram on the right hand side.

 

Patterns within the Triangle:

Let us now take a look at the numbers in the diagonals of the Pascal Triangle.

Diagonals: The first diagonal is consisted of only "$1$"s, and the second diagonal is consisted of our Natural Numbers ($1,2,3$, etc). The third diagonal is consisted of numbers what are commonly referred as  triangular numbers.
This is the Triangular Number Sequence: $1, 3, 6, 10, 15, 21, 28, 36, 45, …$ This sequence not only represent the third diagonal of the Pascal Triangle, but it can also be generated from the same Pascal Triangle in a different way. Count the total number of elements in the Pascal Triangle upto the $i$-th row, that will give us the $i$-th Triangular number in the Triangular Number Sequence.
What about the elements in the fourth diagonal? The sequence $1, 4, 10, 20, ...$, representing the fourth diagonal is entirely consisted of Tetrahedral Numbers. The $n$-th tetrahedral number is the sum of the elements up to the $n$-th triangular number. For example, the $4$-th tetrahedral number $20$ is equal to the sum of the first $4$ triangular numbers, i.e., $1, 3, 6, 10$.
The square of a number in second diagonal is equal to the sum of the number next to it (in the same row) and the number which is below both of these. Both the summands are elements of the third diagonal. For example – 
  • $2^{2} = 1 + 3 = 4$
  • $3^{2} = 3 + 6 = 9$,
  • $4^{2} = 6 + 10 = 16$,
  • $5^{2} = 10 + 15 = 25$, 
Rima got fascinated and then started to look into the numbers of the Pascal Triangle horizontally.

Horizontal Sums: The sum of the elements in a row in Pascal Triangle are equal to powers of $2$. It starts with $2^{0}$, the sum of the elements of the first row, followed by sum of the elements of the second row which is $2$ and so on.

Exponents of 11: Exponents of $11$ can be derived from the Pascal Triangle. The $i$-th row of the Pascal Triangle, can be interpreted to get the value of $11^{i}$. For $i = 0, 1, 2, 3$ and $4$, the digits in the value of $11^{i}$, resembles with the elements of the $i$-th row of the Pascal Triangle.But what happens with $11^{5}$ ? Interpret the elements of the $5$-th row of the Pascal Triangle as follows.
$1.10^{5} + 5.10^{4} + 10.10^{3} + 10.10^{2} + 5.10 + 1$ = $161051 = 11^{5}$. The same thing happens with $11^{6}$ etc. 
Rima was thrilled, now she had so much to share with her sister!! But somehow she couldn’t stop reading.

Fibonacci Sequence: Start with elements of the first diagonal and then move up diagonally taking into account numbers (which were not considered previously) and then add up the values (as illustrated), we will obtain Fibonacci Sequence. The Fibonacci Sequence starts with "$0$, $1$" and then continues by adding the two previous elements of the sequence, for example $3+5=8$, then $5+8=13$, etc.



Pascal triangle is also symmetrical. If we imagine a line cutting the Pascal triangle equally into two halves, then the numbers on the left side have identical matching numbers on the right side. 

Pascal's Triangle is also having interesting application as it can help to visualize in how many ways heads and tails can combine.  For example, if you toss a coin three times, there is only one combination that will give you three heads (HHH), but there are three that 
will give two heads and one tail (HHT, HTH, THH), also three that give one head and two tails (HTT, THT, TTH) and one for all Tails (TTT). This is the pattern "$1,3,3,1$" in Pascal's Triangle. The triangle therefore shows in how many ways Head and Tail can be combinbed. This, therefore, can help us calculate the probability of $k$ successes in an experiment of $n$ statistically independent Bernoulli Trials.

No. of Tosses

Possible Results (Grouped)

Pascal's Triangle

1

H
T

$1, 1$

2

HH
HT TH
TT

$1, 2, 1$

3

HHH
HHT, HTH, THH
HTT, THT, TTH
TTT

$1, 3, 3, 1$

4

HHHH
HHHT, HHTH, HTHH, THHH

HHTT, HTHT, THHT,HTTH,THTH, TTHH
HTTT, THTT, TTHT, TTTH
 TTT

$1, 4, 6, 4, 1$

The value at any place in Pascal's triangle can be computed using the following formula $$C(n, k) =  \frac{n!}{k!.(n – k)!}$$ where $n = 0, 1, 2, 3, ...$ denotes the row number and $k = 0, 1, 2, 3, ...$ denotes the column number within the $n$-th row.  $C(n, k)$ is referred as  "$n$ choose $k$" and can also be written  nCk  or even nCk. The first row with one element is considered as row $0$ and the first element of each row is considered as the element of the $0$-th column. Pascal’s triangle, therefore, can also be referred as “$n$ choose $k$” triangle. The "$!$" in the above formula implies factorial which means to multiply a series of descending natural numbers. For example, $n! = n(n-1)(n-2)….3.2.1.$ Also note that 0! Is defined as equal to 1.
Example: $2$-nd term in the $5$-th row of the Pascal's Triangle is "$10$". The formula helps us in getting it quickly without computing the previous rows of the triangle. Since, $C(5, 2) = 5! / 2!.3! = 120 / 2.6 = 10$.
Rima thought that this is really very useful since she can now work out any value in Pascal's Triangle directly (without calculating the whole triangle above it). Pascal's Triangle can also be used to find out the coefficients in a binomial expansion:

Power

Binomial Expansion

Pascal's Triangle

2

(x + 1)2 = 1x2 + 2x + 1

$1, 2, 1$

3

(x + 1)3 = 1x3 + 3x2 + 3x + 1

$1, 3, 3, 1$

4

(x + 1)4 = 1x4 + 4x3 + 6x2 + 4x + 1

$1, 4, 6, 4, 1$

Now Rima was very excited. She had so much information to share now with her little sister.

About the Author
Sutapa Chakraborty is currently serving as a lecturer of Mathematics for B.Sc (Computer Science) at Indira College of Science & Commerce, Pune. She has a long experience of teaching students of Class $IX - XII$ for more than $15$ years. She has strong interest in the application areas of Mathematics and prefers to teach in a way which students can relate to themselves in their daily life. 


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