The Mathematical Secrets of Pascal’s Triangle

Shweta Katheria

Mar 6, 2019 . 4 min read

This may look like a neatly arranged stack of numbers, but it’s actually a mathematical treasure trove. It is known as the Pascal’s Triangle, after French mathematician Blaise Pascal. So what is it about this that has so intrigued mathematicians the world over? In short, it’s full of patterns and secrets. We will discuss some of the secrets here.

  1. Generation of Pascal Triangle:

pascal

First and foremost, there’s the pattern that generates it. Start with 1 and imagine invisible zeros on either side of it. Add them together in pairs and you will generate next row. Now do that again and again, keep going and you will wind up with something like this. Though really Pascal’s triangle goes on infinitely.

2.Binomial Expansion of the form x+y:

pascal

Now each row corresponds to what’s called the coefficients of a Binomial expansion of the form x plus y raised to the n (x + y)n, where n is the number of the row and we start counting from zero. So if you make n equal to 2 (n=2)and expand it, you get x²+2xy+y². The coefficient or numbers in front of the variables are the same as the numbers in that row of Pascal’s triangles. You will see the same thing with n = 3which expands to this. So the triangle is a quick and easy way to look up all of these coefficients, but there’s much more.

pascal

3. Powers of 2 :

pascal

Source: http://jwilson.coe.uga.edu/EMAT6680Su12/Berryman/6690/BerrymanK-Pascals/BerrymanK-Pascals.html

 

Now let’s take a look at powers of 2. If you notice, the sum of the numbers in Row 0 is 1 or 2^0. Similarly, in Row 1, the sum of the numbers is 1+1 = 2 = 2^1. If you will look at each row down to row 5, you will see that this is true. In fact, if Pascal’s triangle was expanded further past Row 5, you would see that the sum of the numbers of any nth row would equal to 2^n.

4. Primes in Pascal triangle :

pascal

Source: http://jwilson.coe.uga.edu/EMAT6680Su12/Berryman/6690/BerrymanK-Pascals/BerrymanK-Pascals.html

 

When you look at Pascal’s triangle, find the prime numbers that are the first number in the row. That prime number is a divisor of every number in that row. For eg. Look at the diagram at the left in row 3 you can see 2 and in row 4th you can see 3 similarily 5…. 7…. 11…. 13 they all are the divisor in their respective row.

5. Magic 11’s :

pascal

Source: http://jwilson.coe.uga.edu/EMAT6680Su12/Berryman/6690/BerrymanK-Pascals/BerrymanK-Pascals.html

 

Each row represents the numbers in the powers of 11 (carrying over the digit if it is not a single number). For example, the numbers in row 4 are 1, 4, 6, 4, and 1 and 114 is equal to 14,641. Look at row 5. The numbers in row 5 are 1, 5, 10, 10, 5, and 1. Since 10 has two digits, you have to carry over, so you would get 161,051 which is equal to 115.

6. Hockey Stick Pattern :

Well, what’s that hockey stick is here ?

pascal

Source: http://jwilson.coe.uga.edu/EMAT6680Su12/Berryman/6690/BerrymanK-Pascals/BerrymanK-Pascals.html

 

It’s a kind of pattern you see in Pascal’s triangle as you Start with any number in Pascal’s Triangle and proceed down the diagonal. Then change the direction in the diagonal for the last number. That last number is the sum of every other number in the diagonal. As you can see from the figure 1+3+6=10 shown in red and similarly for green hockey stick pattern 1+7+28+84=120.

7. Triangular Number :

Well, what’s the Triangular number?

pascal

Source: http://jwilson.coe.uga.edu/EMAT6680Su12/Berryman/6690/BerrymanK-Pascals/BerrymanK-Pascals.html

 

It is a series of numbers (1, 3, 6, 10, 15, etc.) obtained by continued summation of the natural numbers 1, 2, 3, 4, 5, etc. As a matter of fact, you can obtain those triangular numbers from the Pascal triangle. If you start with row 2 and start with 1, the diagonal contains the triangular numbers in our pascal’s triangle.

8. Squares Numbers :

pascal

Source: http://jwilson.coe.uga.edu/EMAT6680Su12/Berryman/6690/BerrymanK-Pascals/BerrymanK-Pascals.html

 

Down the diagonal, as in the figure left, are the square numbers. You can find them by summing 2 numbers together. This can be done by starting with 0+1=1=12, then 1+3=4=22 , 3+6 = 9=32 and so on.

9. Fibonacci Series :

pascal

Source: http://jwilson.coe.uga.edu/EMAT6680Su12/Berryman/6690/BerrymanK-Pascals/BerrymanK-Pascals.html

 

Well, what is Fibonacci series? It is a series of numbers in which each number ( Fibonacci number) is the sum of the two preceding numbers. The simplest is the series 1, 1, 2, 3, 5, 8, etc. You can get Fibonacci series from Pascal’s triangle too. If you take the sum of the shallow diagonal, you will get the Fibonacci numbers.

Using Pascal’s Triangle:

  1. Probability:

Pascal’s Triangle can show you how many ways heads and tails can combine. This can then show you the probability of any combination. For eg. 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.

pascal

Example: What is the probability of getting exactly two heads with 4 coin tosses?

There are 1+4+6+4+1 = 16 (or 24=16) possible results, and 6 of them give exactly two heads. So the probability is 6/16, or 37.5%

2. Combinations :

The triangle also shows you how many Combinations of objects are possible.

Example: You have 16 pool balls. How many different ways could you choose just 3 of them (ignoring the order that you select them)?

Answer: go down to the start of row 16 (the top row is 0), and then along 3 places (the first place is 0) and the value there is your answer, 560.

Here is an extract at row 16 :

pascal

Well, that’s just the brief introduction and some brief magical patterns you may see in Pascal’s triangle. You can also see many patterns also like that of Fractals, Sierpinski triangle etc. But that’s not the end my friend, it is hiding a lot of secrets within and who knows, maybe you may find one someday. And if you could, let me know. I am a big fan of MATHEMATICS. And you know what MATHS is fun.

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