11 and two..............................Euler's product...................with an S of 1.......for the primes..........begins..................2, 3/2, 5/4, 7/6....................two......could be written as (1 + 1)...........binomials...........binomial expansion........two terms.......a + b...

Pascal's Triangle

One of the most interesting Number Patterns is Pascal's Triangle (named after Blaise Pascal, a famous French Mathematician and Philosopher). To build the triangle, start with "1" at the top, then continue placing numbers below it in a triangular pattern. Each number is the numbers directly above it added together. (Here I have highlighted that 1+3 = 4)

Patterns Within the Triangle

Diagonals

The first diagonal is, of course, just "1"s
The next diagonal has the Counting Numbers (1,2,3, etc).
The third diagonal has the triangular numbers
(The fourth diagonal, not highlighted, has the tetrahedral numbers.)

Symmetrical

The triangle is also symmetrical. The numbers on the left side have identical matching numbers on the right side, like a mirror image.

Horizontal Sums

What do you notice about the horizontal sums?
Is there a pattern?
They double each time (powers of 2).

Exponents of 11

Each line is also the powers (exponents) of 11:

• 11⁰=1 (the first line is just a "1")
• 11¹=11 (the second line is "1" and "1")
• 11²=121 (the third line is "1", "2", "1")
• etc!

But what happens with 11⁵ ? Simple! The digits just overlap, like this:

The same thing happens with 11⁶ etc.

Squares For the second diagonal, the square of a number is equal to the sum of the numbers next to it and below both of those. Examples: 32 = 3 + 6 = 9, 42 = 6 + 10 = 16, 52 = 10 + 15 = 25, ...

There is a good reason, too ... can you think of it? (Hint: 4²=6+10, 6=3+2+1, and 10=4+3+2+1)

Fibonacci Sequence

Try this: make a pattern by going up and then along, then add up the values (as illustrated) ... you will get the Fibonacci Sequence.

(The Fibonacci Sequence starts "0, 1" and then continues by adding the two previous numbers, for example 3+5=8, then 5+8=13, etc)

Odds and Evens

If you color the Odd and Even numbers, you end up with a pattern the same as the Sierpinski Triangle

Using Pascal's Triangle

Heads and Tails

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 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.

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, HTTH, THHT, THTH, TTHH HTTT, THTT, TTHT, TTTH TTTT	1, 4, 6, 4, 1
	... etc ...

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