How Is Pascal's Triangle Used?

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lv1997 | (Level 2) eNoter

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Pascal's triangle is a triangular array of numbers, arranged so that the ends of every row are 1's. Each of the numbers in-between the 1's is equal to the sum of the two numbers above it on either side.

Pascal's Triangle Row 0           1           Row l         1   1         Row 2       1   2   1       Row 3     1   3   3   1     Row 4   1   4   6   4   1   Row 5 1   5   10   10   5  

1

Pascal's triangle is used in order to take a binomial and raise it to a power.

For example: (a+b)^n

Where "n" signifies the number of the row.

For example: (a+b)^5 would be set up as the following:

1a^5 + 5a^4b + 10a^3b^2 + 10a^2b^3 + 5ab^4 +1b^5

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fact-finder | (Level 3) Valedictorian

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Pascal's triangle is a triangular array of numbers, arranged so that the ends of every row are ls. Each of the numbers in-between the 1s is equal to the sum of the two numbers above it. on either side.

Pascal's Triangle
Row 0 1
Row l 1 1
Row 2 1 2 1
Row 3 1 3 3 1
Row 4 1 4 6 4 1
Row 5 1 5 10 10 5 1

Many mathematical patterns can be found in Pascal's triangle. One pattern can be seen by drawing five lines-one through each diagonal. You'll find that the sum of the numbers in each diagonal is equal to the sum of two numbers: the bottom number in the row and the number to the right. For example, the sum of the numbers in the first diagnoal (1+1+1+1+1+1) equals 6. If you add the bottom number in that row (1) and the number to the right (5), you'll also get 6.

Another pattern is that the sum of the numbers in each row is equal to 2, raised to the power of the row number (2row#). When a number is raised to a given power, it is multiplied by itself a given number of times (23 is 2x2x2=8). For example, in Row 0, the sum of the numbers is 1, which is equal to 2°. In Row 1, the sum of the numbers is 2, which is equal to 21; and in Row 2 the sum of the numbers is 4, which is equal to 22.

Pascal's triangle is often used in calculus to solve problems that involve taking the sum of two terms, called a binomial, and raising it to a power. The particular power to which the binomial is raised, (a+b)n, indicates which row of the triangle to use. That particular row contains the coefficients (a factor by which a given number is multiplied) of each term in the expansion of the binomial.

For example, (a + b)1 = la1 + lb1 uses the numbers in Row 1 of the triangle. The equation (a + b)2 = la2 + 2ab + lb2 uses the coefficients in Row 2 of the triangle. You can see the 1, the 2, and the 1, in Row 2. You would use Row 3 to come up with the coefficients in the following: (a + b)3 = la3 + 3a2b + 3ab2 + lb3. (The first line of the triangle correlates to (a + b)0, which is always equal to 1.)

Pascal's triangle was developed by French mathematician and physicist (a scientist specializing in the interaction between matter and energy) Blaise Pascal (1623-1662). It is especially useful in calculating probabilities.

There is some evidence that a triangle similar to Pascal's triangle was used in China around A.D. 1100. The first publication of the triangle was probably in a book called Piling-Up Powers and Unlocking Coefficients by Chinese mathematician Liu Ju-Hsieh.

Sources: Barnhart, Robert K. The American Heritage Dictionary of Science, p. 474; Motz, Lloyd, and Jefferson Hane Weaver. Conquering Mathematics, pp. 53-56; Temple, Robert K.G. The Genius of China, pp. 146-47.

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