Block, SkipCaptcha, apieditor, Automoderated users, Bots, edit, ratelimit-exempt
58,183
edits
Changes
Jump to navigation
Jump to search
Line 1:
Line 1: − + − + Line 13:
Line 13: − + Line 27:
Line 27: − +
clean up & uniformity
[[Image:Pascal.jpg|thumb|289px|right|Pascal's triangle up to row six]]
[[Image:Pascal.jpg|thumb|289px|right|Pascal's triangle up to row six]]
'''Pascal's triangle''' is a triangular arrangement of the coefficients of a binomial expansion. Although it was discovered centuries earlier,<ref>"Napier ... discovered what eventually would be called "Pascal's Triangle" and placed it in common use long before Pascal was even born." [http://history-computer.com/People/NapiersBio.html] </ref> the triangle's full significance was first understood by 17<sup>th</sup> century mathematician and religious scholar [[Blaise Pascal]].<ref>http://pages.csam.montclair.edu/~kazimir/history.html</ref> Pascal's triangle is constructed by starting with a single 1, and then moving downward and out so that each new element is the sum of the two numbers diagonally above it. (Elements outside the triangle are considered to be zero.)
'''Pascal's triangle''' is a triangular arrangement of the coefficients of a binomial expansion. Although it was discovered centuries earlier,<ref>"Napier ... discovered what eventually would be called "Pascal's Triangle" and placed it in common use long before Pascal was even born." [http://history-computer.com/People/NapiersBio.html]</ref> the triangle's full significance was first understood by 17th century mathematician and religious scholar [[Blaise Pascal]].<ref>http://pages.csam.montclair.edu/~kazimir/history.html</ref> Pascal's triangle is constructed by starting with a single 1, and then moving downward and out so that each new element is the sum of the two numbers diagonally above it. (Elements outside the triangle are considered to be zero.)
The top row of the triangle (containing just a single 1) is the 0<sup>th</sup> row, with rows below it being 1<sup>st</sup>, 2<sup>nd</sup>, and so on. The left-most element in each row (which will always be a 1) is the 0<sup>th</sup> element of that row, with the following elements continuing as 1<sup>st</sup>, 2<sup>nd</sup>, and so forth.
The top row of the triangle (containing just a single 1) is the 0th row, with rows below it being 1st, 2nd, and so on. The left-most element in each row (which will always be a 1) is the 0th element of that row, with the following elements continuing as 1st, 2nd, and so forth.
==Applications==
==Applications==
===Combinations===
===Combinations===
For a [[Combinatorics#Combinations|mathematical combination]] of the form nCr, the solution can be found at row n, element r of Pascal's triangle. For the combination 5C3, the solution is the number at the 3<sup>rd</sup> element of the 5<sup>th</sup> row.
For a [[Combinatorics#Combinations|mathematical combination]] of the form nCr, the solution can be found at row n, element r of Pascal's triangle. For the combination 5C3, the solution is the number at the 3rd element of the 5th row.
<math>5C3 = 10</math>
<math>5C3 = 10</math>
:<math> = \frac{n!}{r!(n-r)!} = {n \choose r}.</math>
:<math> = \frac{n!}{r!(n-r)!} = {n \choose r}.</math>
==Further Reading==
==Further reading==
<small>
<small>
*[http://ptri1.tripod.com/ All You Ever Wanted to Know About Pascal's Triangle and more]
*[http://ptri1.tripod.com/ All You Ever Wanted to Know About Pascal's Triangle and more]