Difference between revisions of "Pascal's triangle"
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| − | ''' | + | '''Pascal's triangle''' is a triangular arrangement of the coefficients of a binomial expansion. Although the it was discovered in the eleventh century, 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.) |
| − | + | The top row of the triangle (containing just a single 1) is the 0<sup>th</sup> row, with rows below it 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 ( | ||
==Applications== | ==Applications== | ||
| − | Pascal's triangle has | + | Pascal's triangle has numerous applications and properties, some of which are beyond the scope of this article.<ref>http://ptri1.tripod.com/</ref> Some are useful, others are merely interesting. Only a few of the more well-known applications of Pascal's triangle are described below. |
===Binomial Expansion=== | ===Binomial Expansion=== | ||
| + | For the binomial <math>(a+b)^n</math>, the coefficients of the expansion are found in the n<sup>th</sup> row of Pascal's triangle. (Remember the top row is row 0.) For example, the binomial coefficients for <math>(a+b)^4</math> are the numbers in row 4 of the triangle. | ||
| + | <math>(a+b)^4=1a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + 1b^4</math> | ||
| + | |||
| + | ===Combinations=== | ||
| + | For a [[Combination|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. | ||
| + | <math>5C3 = 10</math> | ||
| + | |||
| + | ==Further Reading== | ||
| + | <small> | ||
| + | *[http://ptri1.tripod.com/ All You Ever Wanted to Know About Pascal's Triangle and more] | ||
| + | </small> | ||
| + | |||
| + | ==References== | ||
| + | <references/> | ||
| + | |||
| + | [[Category:Mathematics]] | ||
Revision as of 00:41, June 3, 2010
Pascal's triangle is a triangular arrangement of the coefficients of a binomial expansion. Although the it was discovered in the eleventh century, the triangle's full significance was first understood by 17th century mathematician and religious scholar Blaise Pascal.[1] 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 0th row, with rows below it 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
Pascal's triangle has numerous applications and properties, some of which are beyond the scope of this article.[2] Some are useful, others are merely interesting. Only a few of the more well-known applications of Pascal's triangle are described below.
Binomial Expansion
For the binomial , the coefficients of the expansion are found in the nth row of Pascal's triangle. (Remember the top row is row 0.) For example, the binomial coefficients for are the numbers in row 4 of the triangle.
Combinations
For a 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.
Further Reading