Mathematics
The Peculiarity of Pascal’s Triangle
Innumerable Applications and Properties
Pascal’s Triangle is one of the most important treasure troves for all the mathematicians since centuries. But what’s so special about it? Isn’t it just a stack of numbers formed by the addition arranged in triangular pattern? Well no! It is much more than just addition. Let’s fully understand Pascal’s triangle.
Pascal’s triangle is actually not discovered by Pascal as the name suggests. The work on Pascal’s triangle started way earlier, but the 17th-century French mathematician, Blaise Pascal, had major contribution. Pascal’s triangle was known by different names in different countries. Indian mathematicians called it the Staircase of Mount Meru. Iranians called is the Khayyam’s triangle. Chinese people called it the Yang Hui’s Triangle. But most of the western world knew it as Pascal’s triangle. Now, in almost every part of the world, Pascal’s triangle is widely used.
Construction of Pascal’s triangle:
To build the triangle, start with “1” at the top, On the next row write two 1’s, forming a triangle. On each subsequent row start and end with 1’s and compute each interior term by summing the two numbers above it. By repeating this process we will end up getting a Pascal triangle and it can go infinitely.
From the animation above, you can clearly see how a Pascal’s triangle is formed and you can extend it to whatever number of rows.
Let’s dive into the applications of Pascal’s triangle!
- Binomial Expansion: Pascal’s triangle helps to determine the coefficients which arise during binomial expansion
As you can see from the table above, the coefficients in the expansion of (a+b)^n can be found in the nth row of the Pascal’s triangle.
2. Probability: Pascal’s triangle is used in probability. It gives us the number of combinations of heads or tails that are possible from the number of tosses, giving the probability of the combination.
3. Combination: Another useful application of Pascal’s triangle is in the calculation of combinations. We know that If the number of combinations of n things taken k at a time (called n choose k) can be found by the equation C(n, k) = (n k) {in vertical} = n!/[k!*(n-k)!].
But this is a formula for an entry of a cell in Pascal’s triangle as well. So, we can just lookup that particular entry and in the triangle and find it out.
Note: For doing this, the numbering of first row and the first entry in the row should start from 0.
Now, let’s dive into patterns and properties of Pascal’s triangle:
- Each number is the sum of pair of numbers above it.
- The triangle is symmetrical. The triangle remains the same even if we revert the numbers from right to left.
- Adding up the numbers in each row gives the power of 2.
4. Reading the numbers from left to right gives the power of 11.
5. Summing up the numbers in the diagonals end up forming the Fibonacci Sequence.
6. The edge diagonals are always 1’s. The first diagonals next to edge diagonals are natural numbers. The second diagonals are triangular numbers. Similarly, the next diagonals are tetrahedral numbers.
A triangular number is a number that can make an equilateral triangular dot pattern. For example, 1, 3, 6, 10 and 15 are triangular numbers.
Similarly, a tetrahedral number is a number that can make a tetrahedral or triangular pyramid pattern. For example, 1, 4, 10 and 20 are tetrahedral numbers.
7. Shading all the even numbers will give a fractal. This is also the recursive of Sierpinski Triangle.
8. The product of alternate three numbers in the hexagon is equal to the product of the remaining three alternate numbers.
Let’s take see the examples below to make it more clear.
As you can see, the product of alternate three numbers in the hexagon is equal to the product of the remaining three alternate numbers.
There are other patterns and properties such as Hockey Stick Pattern, Catalan numbers, and many more associated with the Pascal’s Number. I have left those as the exercise for the readers to find out and research more.
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