Ken Ward's Mathematics Pages

Series - Pascal's Triangle

Series Contents

Page Contents

1. Pascal's Triangle
2. Formula for the Binomial Coefficient (Numbers in Pascal's Triangle)
3. Formula for the Binomial Coefficients  (Pascal's Observation)
4. Relationships Between Coefficients
1. Symmetry Law
2. Upper Summation
3. Parallel Summation
5. Natural Numbers, Again

Pascal's Triangle

Blaise Pascal (1623-1662) is associated with the triangle of numbers which bears his name, although it is known as Tartaglio's Triangle in Italy, and was known at least 700 years before Pascal by Indian, Chinese, and other mathematicians, perhaps a long time before that too.

Our interest here is with the Binomial Theorem. The Theorem deals with equations of the form:

[1.1]
That is the expansion of factors consisting of 2 items (in this case a and b) raised to a power (Other factors can be included, which we will mention later).

In studying these equations, we can note that the sum of the powers in each term is equal to the power on the left-hand side. We can write the coefficients in a table for further study:

Table 1: Pascal's Triangle

Pascal's Triangle
	k
n	0	1	2	3	4	5	6	7	8	9	10
10	1	10	45	120	210	252	210	120	45	10	1
9	1	9	36	84	126	126	84	36	9	1
8	1	8	28	56	70	56	28	8	1
7	1	7	21	35	35	21	7	1
6	1	6	15	20	15	6	1
5	1	5	10	10	5	1
4	1	4	6	4	1
3	1	3	3	1
2	1	2	1
1	1	1
0	1

In the table, n is the power, k is the term number. We fill the table in from the values of the equations (The blank cells are zero).

We call the numbers in the triangle Binomial Coefficients and represent them by , where n refers to the n-position in the column and k refers to the row position. At the moment, this symbol is defined as the n, k position in the table. When n and k are positive integers (or zero), this symbol is, in fact, a combination, but it is always a Binomial Coefficient, and only sometimes a combination. It is read as "n over k". At the moment, however, we do not know what it is apart from the given definition.

A careful study of the table, reveals that any number is made up of the sum of the number above it and the one to the left. Using our symbol:
[1.2, the Addition Law]
Where n is the row number and k is the column number in the table above.

Of course, we haven't proved this yet.

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Formula for the Binomial Coefficients (Numbers in Pascal's Triangle)

By studying the table, we note that
[1.3]
[1.4]

It is also evident that when k>n, and n and k are nonnegative integers, then the term is zero:
[1.5]
(This is why, when n and k are nonnegative integers, the Binomial Series seems to be finite, because later terms are zero)

Looking at term 2, we can note that each number is a sum of natural numbers.  And
[1.6]
So,
[1.7]

I do not recognise the numbers in the term 3's.

[Actually the formula is: , revealing the Upper Summation Formula, which we will consider later]

Therefore, I collect more terms, and expand Pascal's Triangle thus:

k
n	0	1	2	3	4	5	6	7
0	1
1	1	1
2	1	2	1
3	1	3	3	1
4	1	4	6	4	1
5	1	5	10	10	5	1
6	1	6	15	20	15	6	1
7	1	7	21	35	35	21	7	1
8	1	8	28	56	70	56	28	8
9	1	9	36	84	126	126	84	36
10	1	10	45	120	210	252	210	120
11	1	11	55	165	330	462	462	330

We can use the method of differences to find out what the formula is. For term 3, we create the difference table:

n	0	1	2	3	4	5	6
Series	0	1	4	10	20	35	56
Δ1		1	3	6	10	15	21
Δ2			2	3	4	5	6
Δ3				1	1	1	1

Because the Δ3 terms are all constants, we know the equation is a cubic, and can be represented as:
ax³+bx²+cx+d

Through making simultaneous equations, we find that equation is:
[1.8]

However, this is the sum of n terms, and term 3 is a sum to n-2, which gives:
[1.9]

In this way, we can derive formulae for the terms of Pascal's Triangle, and hence the Binomial Coefficients.

Formula for the Binomial Coefficients  (Pascal's Observation)

 [3.3] The table below was called the arithmetic triangle.
Table 2 (Arithmetic Triangle)

n
		0	1	2	3	4	5	6	7	8
k	0	1	1	1	1	1	1	1	1	1
	1	1	2	3	4	5	6	7	8
	2	1	3	6	10	15	21	28
	3	1	4	10	20	35	56
	4	1	5	15	35	70
	5	1	6	21	56
	6	1	7	28
	7	1	8
	8	1

How the table is created

The k values relate to the row, and the n values relate to the start of a diagonal. These numbers correspond to Pascal's Triangle.
We make this table by writing down a series of 1's, which is the series of constants.

Next we simply add these, so the numbers in the row below, 1, 2, ... are the sum of the first row, up to the given position. So, 4 in row 2, is the sum of the 1's up to 4's position. These are, of course, the natural numbers.

The next row, 1, 3, 6.. is composed of the sums of the previous row, and each cell is a part sum of the natural numbers.

In a similar manner each row is the part sums of the previous row.

Our familiar Binomial Coefficients appear on the diagonals. So the first 1 represents the Binomial Expansion of power zero (n=0). The next diagonal, 1, 1 is the expansion when n=1.  And the last one is the expansion when n=8.

So any diagonal represent the Binomial Coefficients of successive n's.

Pascal's Observation

For each successive number on the diagonal, the lower relates to the upper as horizontal distance to the left of the lower to the beginning of the diagonal relates to the horizontal distance to the right of the upper to the beginning of the diagonal, both inclusive.

For instance, so 35 (the lower) relates to 21 (the upper) as 5 relates to 3.

What this does is to give us a way of relating successive Binomial Coefficients.

Modern Formulation

Using the notation for our regular Pascal's Triangle:
[1.3.1]
Where n is the power and k is the term number.

That is, we can find a formula to find the next term in an expansion by rearranging 1.3.1:
[1.3.2]

Finding the Values of the Coefficients (Numbers in Pascal's Triangle)

From Pascal's Triangle, we have noted that the coefficient of term 0 is 1, and the coefficient of term 2 is n. We have calculated terms 2 and 3, and might have an idea what the general formula for the numbers might be, but would need to do some series algebra to collect more information.

Now we note Pascal's observation, we can calculate the values quite easily in a recursive manner.
[1.3.3]

The formula seem to fly out of the equation, and we can easily generalise:
[1.3.4]

The factorials become very clear. So we can confidently say:

And, because substituting (n-k) for k gives the same result, we can say:
[1.3.5, Symmetry Law]

Symmetry is evident by observing that in Pascal's Triangle, the kth coefficient is equal to the (n-k)th coefficient.

Of course, n is a nonnegative integer (non-negative).

Relationships Between Coefficients

Table 2 is reproduced from above, and is the Arithmetic Triangle. The rows are made up of sums from the row above.

As mentioned before, the n and k values relate to Pascal's Triangle: in Table 2, the k's relate to the rows, but the n's relate to the diagonals.

Table 2 (Arithmetic Triangle)
		n
		0	1	2	3	4	5	6	7	8
k	0	1	1	1	1	1	1	1	1	1
	1	1	2	3	4	5	6	7	8
	2	1	3	6	10	15	21	28
	3	1	4	10	20	35	56
	4	1	5	15	35	70
	5	1	6	21	56
	6	1	7	28
	7	1	8
	8	1

Upper Summation

Each row relates to the respective n over k terms. For instance, in row 2, the coefficient, 10, is the 5 over 2 term in Pascal's Triangle. It is a sum of part of the upper row, so:
[2.1.1]
Keeping with the coefficient, 10, we note that 5 over 2 is the sum of the first 4 terms from 1 to (n-1), n=5.

In general, we find:
 [2.1.2]

And by writing n+1 for n:
[2.1.3]

Parallel Summation

In a similar fashion as before, we can note that the sums of numbers in Pascal's Triangle are the numbers just above the last number in the diagonal.

Pascal's Triangle
	k
n	0	1	2	3	4	5	6	7
12	1	12	66	220	495	792	924	792
11	1	11	55	165	330	462	462	330
10	1	10	45	120	210	252	210	120
9	1	9	36	84	126	126	84	36
8	1	8	28	56	70	56	28	8
7	1	7	21	35	35	21	7	1
6	1	6	15	20	15	6	1
5	1	5	10	10	5	1
4	1	4	6	4	1
3	1	3	3	1
2	1	2	1
1	1	1
0	1

For instance, the highlighted yellow rows have their sum as red square above highlighted in red.
[2.2.1]

The general formula is:
[2.2.2, Parallel Summation]
This is the sum of Binomial Coefficients which increase in value in a parallel manner, so the distance between the upper and lower indices is constant.

Natural Numbers, Again

The equation below shows the sum of the natural numbers as the sum of Binomial Coefficients:
[3.1.1]

From 2.1.3, we know that the right-hand sum is:
[3.1.2]
Giving us the sum of the first n natural numbers.

Here the basic expression used above is simply:
[3.1.3]

However, it isn't that hard to figure out:
[3.1.4]

And we more or less just write down:
[3.1.5]

Giving us the sum of the squares of the natural numbers:
[3.1.6]










Ken Ward's Mathematics Pages

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Faster Arithmetic - by Ken Ward