Pascal's triangle - Revision history

FredericBernard: /* Binomial Expansion */ Wikify

2017-09-12T14:07:54Z

‎Binomial Expansion: Wikify

DavidB4-bot: clean up & uniformity

2016-07-13T17:15:55Z

clean up & uniformity

Ed Poor: not sure of the exact century: see Napier

2012-07-20T14:36:45Z

not sure of the exact century: see Napier

ReligiousRight: /* Applications */

2010-06-03T19:51:54Z

‎Applications

ChrisGT90: Picture added

2010-06-03T17:06:44Z

Picture added

ChrisGT90: /* Combinations */ Fixed link for combinations

2010-06-03T01:57:05Z

‎Combinations: Fixed link for combinations

ChrisGT90: typo fix

2010-06-03T00:43:10Z

typo fix

ChrisGT90: Completion of article. Hopefully more will be added.

2010-06-03T00:41:50Z

Completion of article. Hopefully more will be added.

ChrisGT90: Created page with ''''Saved part-way through to avoid any accidental loss of work. This article will hopefully be completed within the half hour.''' '''Pascal's triangle''' is a triangular arrange…'

2010-06-03T00:02:47Z

Created page with ''''Saved part-way through to avoid any accidental loss of work. This article will hopefully be completed within the half hour.''' '''Pascal's triangle''' is a triangular arrange…'

New page

'''Saved part-way through to avoid any accidental loss of work. This article will hopefully be completed within the half hour.'''

'''Pascal's triangle''' is a triangular arrangement of the coefficients of a binomial expansion. It is constructed by placing a single 1, and then moving downward and out so that each number is the sum of the two numbers directly above it. (Elements outside the triangle are considered to be zero.)

The top row of the triangle (with just a 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 many applications and properties. Some are useful, others are merely interesting. Only a few of the applications of Pascal's triangle are described below.

===Binomial Expansion===

@@ Line 9: / Line 9: @@
 ===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.
+For the [[binomial expansion]] of <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>

@@ Line 1: / Line 1: @@
 [[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==
@@ Line 13: / Line 13: @@
 ===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>
@@ Line 27: / Line 27: @@
 :<math> = \frac{n!}{r!(n-r)!} = {n \choose r}.</math>
-==Further Reading==
+==Further reading==
 <small>
 *[http://ptri1.tripod.com/ All You Ever Wanted to Know About Pascal's Triangle and more]

@@ Line 1: / Line 1: @@
 [[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 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.)
+'''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.)
 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.

@@ Line 15: / Line 15: @@
 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.
 <math>5C3 = 10</math>
 ==Further Reading==

← Older revision		Revision as of 17:06, June 3, 2010
Line 1:		Line 1:
−	'''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.)	+	[[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 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 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 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.

@@ Line 1: / Line 1: @@
-'''Saved part-way through to avoid any accidental loss of work. This article will hopefully be completed within the half hour.'''
+'''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.)
-'''Pascal's triangle''' is a triangular arrangement of the coefficients of a binomial expansion. It is constructed by placing a single 1, and then moving downward and out so that each number is the sum of the two numbers directly 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 (with just a 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.
 ==Applications==
-Pascal's triangle has many applications and properties. Some are useful, others are merely interesting. Only a few of the applications of Pascal's triangle are described below.
+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===