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Sunday, October 28, 2007

Applications of Pascal's Triangle

application of pascal triangle

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(x-1)^2 = x^2 +2*x+1 (1 2 1)
(x-1)^4 = x^4 +4*x^3+6*x^2+4*x + 1 (1 4 6 4 1)

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 just the two numbers 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, and 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.)

Odds and Evens

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

Horizontal Sums

What do you notice about the horizontal sums? Is there a pattern? Isn't it amazing!

It doubles each time (powers of 2).

Fibonacci Sequence

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

(The Fibonacci Sequence is made by adding the two previous numbers, for example 3+5=8, then 5+8=13, etc)

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 odds" (or 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 ...

What is the probability of getting exactly two heads with 4 coin tosses?

There are 1+4+6+4+1 = 16 (or 4×4=16) possible results, and 6 of them give exactly two heads. So the probability is 6/16, or 37.5%

Ref :www.mathsisfun.com/pascals-triangle.html

4 comments:

Anonymous said...

i love math bitch

Anonymous said...

love math bitch

kitty<3 said...

cool! this was helpful! :)

Anonymous said...

ya its relly wonderful technique.....