Difference between revisions of "Pascal's triangle"

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Pascal's triangle is a triangle in which every number is the sum of the two numbers directly above it. The first few rows of Pascal's Triangle are depicted above.  It can be noted that the numbers in Pascal's triangles are the [[binomial coefficients]].  Consequently, Pascal's triangle has many interseting uses and applications.
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Pascal's triangle is a triangle in which every number is the sum of the two numbers directly above
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== Definition ==
  
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Pascal's triangle is a triangle in which every number is the sum of the two numbers directly above it. The first few rows of Pascal's Triangle are depicted above. 
 
[[Image:Pascal.hex2.gif|left|thumb|400px|''[[Pascal's Triangle]]'' ]]
 
[[Image:Pascal.hex2.gif|left|thumb|400px|''[[Pascal's Triangle]]'' ]]
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== Uses ==
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Pascal's triangle has many interseting uses and applications.
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=== Binomial Coefficients ===
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It can be noted that the numbers in Pascal's triangles are the [[binomial coefficients]].  This means that every number in Pascal's Triangle can be expressed as <math>\displaystyle \binom{n}{k-1}</math>, where n is the row number, and k is the place in the row.  For example, <math>\displaystyle \binom{5}{3}</math> would represent the 4th number in the 5th row.  It should be noted here that the first row of Pascal's Triangle is considered to be the row 1, 1, meaning that the "[[apex]]" of Pascal's Triangle is considered "Row 0."
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=== Counting & Probability ===
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If we flip x coins, and want to know what the probability is that y of them are heads, then we go to the xth row, and find the (y+1)th number (call this a).  Then, we find the sum of the numbers in the xth row (call this b).  Then, the probability is simply <math>\frac{a}{b}</math>

Revision as of 13:59, 23 June 2006

Pascal's triangle is a triangle in which every number is the sum of the two numbers directly above

Definition

Pascal's triangle is a triangle in which every number is the sum of the two numbers directly above it. The first few rows of Pascal's Triangle are depicted above.

Uses

Pascal's triangle has many interseting uses and applications.

Binomial Coefficients

It can be noted that the numbers in Pascal's triangles are the binomial coefficients. This means that every number in Pascal's Triangle can be expressed as $\displaystyle \binom{n}{k-1}$, where n is the row number, and k is the place in the row. For example, $\displaystyle \binom{5}{3}$ would represent the 4th number in the 5th row. It should be noted here that the first row of Pascal's Triangle is considered to be the row 1, 1, meaning that the "apex" of Pascal's Triangle is considered "Row 0."

Counting & Probability

If we flip x coins, and want to know what the probability is that y of them are heads, then we go to the xth row, and find the (y+1)th number (call this a). Then, we find the sum of the numbers in the xth row (call this b). Then, the probability is simply $\frac{a}{b}$

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