application of pascal triangle====================
(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)
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.
Patterns Within the Triangle
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
What do you notice about the horizontal sums? Is there a pattern? Isn't it amazing!
It doubles each time (powers of 2).
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|
|2|| HH |
|1, 2, 1|
|3|| HHH |
HHT, HTH, THH
HTT, THT, TTH
|1, 3, 3, 1|
|4|| HHHH |
HHHT, HHTH, HTHH, THHH
HHTT, HTHT, HTTH, THHT, THTH, TTHH
HTTT, THTT, TTHT, TTTH
|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%