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Pascal's Triangle 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 The first six rows of Pascal's triangle In mathematics, Pascal's triangle is a geometric arrangement of the binomial coefficients in a triangle. It is named after Blaise Pascal. In simple terms, Pascal's triangle can be constructed in the following manner. On the first row, write only the number 1. Then, to construct the elements of following rows, add the number directly above and to the left (if any) and the number directly above and to the right (if any) to find the new value. For example, the numbers 1 and 3 in the fourth row are added to produce 4 in the fifth row. More formally, this construction is using Pascal's identity, which states that -
for positive integers n and k where n ≥ k and with the initial condition -
Pascal's triangle generalizes readily into higher dimensions. The three-dimensional version is called Pascal's pyramid or Pascal's tetrahedron. A higher-dimensional analogue is generically called a "Pascal's simplex". See also pyramid, tetrahedron, and simplex. The triangle Here are 14 lines of Pascal's triangle 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 1 8 28 56 70 56 28 8 1 1 9 36 84 126 126 84 36 9 1 1 10 45 120 210 252 210 120 45 10 1 1 11 55 165 330 462 462 330 165 55 11 1 1 12 66 220 495 792 924 792 495 220 66 12 1 1 13 78 286 715 1287 1716 1716 1287 715 286 78 13 1 Uses of Pascal's triangle Pascal's triangle has many uses in binomial expansions. For example - (X + 1)2 = 1X2 + 2X + 12.
Notice the coefficients are the third row of Pascal's Triangle - 1,2,1. In general, when a binomial is raised to a positive integer power we have: - (X + Y)n = a0Xn + a1Xn-1Y + a2Xn-2Y2 + … + an-1XYn-1 + anYn,
where the coefficients ai in this expansion are precisely the numbers on the (n+1)-th row of Pascal's triangle; in other words, - .
Also, when a Pascal's Triangle is constructed with 2n levels and all odd numbers are shaded, the result is an approximation to the Sierpinski triangle. Shading all multiples of 3, 4, etc. results in other patterns. Properties of Pascal's triangle Some simple patterns are immediately apparent in Pascal's triangle: - The diagonals going along the left and right edges contain only 1s.
- The diagonals next to the edge diagonals contain the natural numbers in order.
- Moving inwards, the next pair of diagonals contain the triangle numbers in order.
There are also more surprising, subtle patterns. From a single element of the triangle, a more shallow diagonal line can be formed by continually moving one element to the left, then one element to the top-left, or by going in the opposite direction. One such example is the line with elements 1,6,5,1, which starts from the row, 1,3,3,1 and ends three rows down. Such a "diagonal" has a sum that is a Fibonacci number. In our example's case, 13. Observe: 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 1 8 28 56 70 56 28 8 1 1 9 36 84 126 126 84 36 9 1 1 10 45 120 210 252 210 120 45 10 1 1 11 55 165 330 462 462 330 165 55 11 1 1 12 66 220 495 792 924 792 495 220 66 12 1 1 13 78 286 715 1287 1716 1716 1287 715 286 78 13 1 The second highlighted diagonal has a sum of 233. In addition, the sum of the squares of the elements of the nth row equals the middle element of the (2n - 1)th. For example, . In general form, that is: -
Another interesting pattern is that on any nth row where n is odd, the middle term minus the term two spots to the left equals a Catalan number, specifically the (n+1)/2 Catalan number. For example: on the 5th row, 6 - 1 = 5, which is the 3rd Catalan number, and (5 + 1)/2 = 3. Also, the sum of the elements of the nth row is equal to 2n-1. For example, the sum of the 5th row is , which is equal to . History The first reference to this triangle occurs in Pingala's book on Sanskrit poetics that may be as early as 450 BC as Meru-prastaara, the "staircase of Mount Meru". The commentators of this book were also aware that the shallow diagonals of the triangle sum to the Fibonacci numbers. It was known to Chinese and Islamic scholars in medieval times. It is said that the triangle was called "Yang Hui's triangle" by the Chinese. Several theorems related to the triangle were known, including the binomial theorem. In Italy, it is referred to as "Tartaglia's Triangle", named for the Italian algebraist Niccolo Fontana Tartaglia who lived a century before Pascal; Tartaglia is credited with the general formula for solving cubic polynomials. In modern times, Pascal's triangle takes its name from the Trait du triangle arithmtique (1655) by Blaise Pascal. In that work, Pascal collected several results then known about the triangle, and employed them to solve problems in probability theory. The "arithmetical" triangle was later called after Pascal by Pierre Raymond de Montmort (1708) and Abraham de Moivre (1730). External links
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