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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

pascals triangle 1s, counting, triangular

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 11⁶etc.

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 6/16, or 37.5%

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:

1    14    91    364  ...
1    15    105   455   1365  ...
1    16   120   560   1820  4368  ...

A Formula for Any Entry in The Triangle

In fact there is a formula from Combinations for working out the value at any place in Pascal's triangle:

It is commonly called "n choose k" and written like this:

Notation: "n choose k" can also be written C(n,k), ⁿC_k or even _nC_k.

The "!" is "factorial" and means to multiply a series of descending natural numbers. Examples:

4! = 4 × 3 × 2 × 1 = 24
7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5040
1! = 1

So Pascal's Triangle could also be
an "n choose k" triangle like this one.

(Note how the top row is row zero
and also the leftmost column is zero)

Example: Row 4, term 2 in Pascal's Triangle is "6" ...

... let's see if the formula works:

Yes, it works! Try another value for yourself.

This can be very useful ... you can now work out any value in Pascal's Triangle directly (without calculating the whole triangle above it).

Polynomials

Pascal's Triangle can also show you the coefficients in binomial expansion:

Power	Binomial Expansion	Pascal's Triangle
2	(x + 1)² = 1x² + 2x + 1	1, 2, 1
3	(x + 1)³ = 1x³ + 3x² + 3x + 1	1, 3, 3, 1
4	(x + 1)⁴ = 1x⁴ + 4x³ + 6x² + 4x + 1	1, 4, 6, 4, 1
	... etc ...

The First 15 Lines

For reference, I have included row 0 to 14 of Pascal's Triangle

126

120

210

252

210

120

165

330

462

330

165

220

495

792

924

792

495

220

286

715

1287

1716