Pascal's triangle has so many strange characteristics... For example, it is connected to other areas of mathematics, like the Fibonacci sequence, and it is used in music theory to calculate the number of ways to arrange a set of notes!
Let's look at Pascal's triangle, how to construct one and its relevance in binomial expansions.
What is Pascal's triangle?
Pascal's triangle is a triangular array of numbers named after the French mathematician Blaise Pascal, where each number is the sum of the two numbers above it. The first row of the triangle is always the number 1, and the second row has two 1s. To form the next row, each adjacent pair of numbers from the row above are added together, with a 1 placed at the beginning and end of the row. This process is repeated to form as many rows as needed.
An illustration of the Pascal's triangle
The diagram above shows the first 8 rows of Pascal's Triangle only, but this can be carried out until infinity. Each row corresponds to a number for n, with the first row being for n = 0.
Pascal's triangle with respective n values
One of the most well-known applications of Pascal's triangle is solving binomial coefficients.
Pascal's triangle and binomial expansions
Binomial coefficients are relevant in the context of binomial expansions.
A binomial expansion refers to the process of finding the power of a binomial expression, such as \((x+y)^n\), where x and y are constants and n is a positive integer. The expansion results in a polynomial expression with \(n+1\) terms. The terms in the expansion can be calculated using the binomial coefficient formula, which involves combinations of the powers of x and y.
A binomial coefficient, \(\binom{n}{k}\), is the number of ways to choose k objects from a set of n distinct objects, regardless of its orders.
The general formula for a binomial expansion is:
\[(x+y)^n = \sum_{k=0}^n \binom{n}{k} x^{n-k}y^k = \sum^n_{k=0} \binom{n}{k} x^k y^{n-k}\]
The binomial coefficients of binomial expansions can be found by using this formula:
\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]
However, Pascal's triangle is the array of binomial coefficients, starting at \(n = 0\) at the very top, so Pascal's triangle can be used to find binomial coefficients.
Carrying out binomial expansion using Pascal's triangle
As mentioned before, Pascal's triangle is a helpful way to determine the binomial coefficients in a binomial expansion.
Let's look at how to expand \((3x+1)^5\).
First, we need to determine n, which is the exponent so in this case 5. This tells us that we will need to construct Pascal's Triangle until row 6 where n = 5. Using the method described above, we get:
An illustration on the use of Pascal's Triangle in binomial expansion
This means we will be using the binomial coefficients 1, 5, 10, 10, 5 and
Plugging this into the binomial formula, we get:
\((3x+1)^5 = 1(3x)^5(1)^0 + 5(3x)^4(1)^1 + 10(3x)^3(1)^2+10(3x)^2(1)^3+5(3x)^1(1)^4+1(3x)^0(1)^5\)
\((3x+1)^5 = 3^5x^5 + 5\cdot 3^4x^4+ 10 \cdot 3^3x^3+10\cdot 3^2x^2+5\cdot 3x+1\)
Which can be simplified to:
\((3x+1)^5 = 243x^5 + 405x^4+270x^3+90x^2+15x+1\)
Pascal's triangle patterns
Pascal's triangle has a specific pattern which makes it easier to construct rather than remember it by heart.
As you might have noticed from the diagram above, each row starts and ends with 1 and the number of elements in each row increases by 1 each time. The number of elements (m) in each row is given by \(m = n + 1\). So the 7th row (n = 6) has 7 elements (1, 6, 15, 20, 15, 6, 1). An element can be found by adding together the two elements above it.
For example, for the third row (n = 2), the 2 comes from adding 1 + 1 from the row above:
Steps in constructing Pascal's Triangle
For the fourth row (n = 3), the two 3s come from adding 1 + 2 from above:
In the fourth row (n = 3) we add 1 + 3 to get 4, 3 + 3 to get 6 and 3 + 1 to get 4:
This process can be repeated as many times as needed until the row we need is reached.
Sum of the rows in Pascal's triangle
In each row, the number obtained by summing all the elements in the row is given by ![]()
. For example for row 3 (n = 2), the sum of the elements is 1 + 2 + 1 = 4 or ![]()
= 4. This is useful to help us work out the sum of the elements for very big rows without having to construct Pascal's triangle For example, we know that for the 20th row (n = 19), the sum would be![]()
The Fibonacci sequence in Pascal's triangle
The Fibonacci series can be found in Pascal's triangle by adding numbers diagonally.
An illustration of the Fibonacci sequence
Pascal's Triangle - Key takeaways
Pascal's triangle can be constructed to help us find binomial coefficients.
It starts at row 1, with n = 0 and a single element, 1.
In each row, the number of elements increases by 1 and is given by \(m = n + 1\), where m is the number of elements.
Each row has a 1 on both extremes and the middle values are found by adding the numbers above.
The sum of each row is \(2^n\).
Fibonacci's sequence can be found by adding the elements diagonally.
We can use Pascal's Triangle to find binomial coefficients and solve binomial expansions of the form \((x+y)^n\).
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