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Pascal’s triangle, named for French philosopher and mathematician Blaise Pascal, is an array of binomial coefficients presented in a triangle form.

Rows of Pascal’s triangle are structured from the top row (0th row) with conventional numerators beginning with 1. Each adjacent row consists of staggered values comprised of the sum of the numbers directly above the value. As rows are constructed below the preceding row, values are applied left to right, with a blank value treated as zero (making the left and right-most values therefore always equal to the top row.

Concepts of Pascal’s triangle have been __known for centuries__, through the work of multiple mathematicians, possibly dating back as early as the 2nd century BC. Pascal presented the earliest formal treatise devoted specifically to the triangle. In his “Treatise on Arithmetical Triangle” Pascal not only presented results based on the triangle’s properties, but also employed them for solving problems in probability theory.

The triangle’s values and formula determine coefficients for binomial expansions.

With the structure provided and formulas for determining the values within the rows, the triangle can be utilized in multiple ways:

Combinations – the Pascal triangle calculator formula can be utilized to solve the equation of an *n* value of items referenced by *k* at a time (stated as n choose k). Taking this combination for the row and entry in the triangle will solve the equation.

Binomial expansion – the formula of expanding powers of binomials can be utilized to populate the values in Pascal’s triangle. If you have a binomial and need to raise the power to perhaps the third or fourth power, rather than performing the multiplications needed, use Pascal’s triangle as your binomial expansion calculator.

Mathematical use of Pascal’s triangle can yield many useful functions:

Probabilities – want to know the probability of getting a certain result from something as basic as a coin toss? Pascal’s triangle can be utilized to solve the instance of any certain result given x number of tosses.

Need to know how many ways you can organize a team into 8 players, given a roster of 10 athletes? Go to row 10 of the triangle, and in to the 8th position (again recalling that the first position is designated as position 0).

With the complexity and functional values of utilizing Pascal’s triangle, you might surmise that there are tools available to utilize the triangle effectively on a scientific or graphing calculator.

To be sure, such functionality has been included – once you know how to get there. In the case of many TI calculators, for example, these functions are located under the MATH menu, (selecting “PRB”).

Following the example as above for “n choose k”, enter the statement as “x nCr 0” (where x = the row to be addressed) and press enter. The statement equates to “x choose 0”. Remember that the first row in the triangle is row zero, and the first entry in any given row is “1”. Using this statement with varying values allows you to access the values in Pascal’s triangle for any given row or position.

This makes calculator functions very powerful, in that you need not construct all levels of the triangle to arrive at a solution – the values in the triangle can be accessed directly. A simple expression such as “30 nCr 6” can, therefore, produce the value in Pascal’s triangle row 30 position 6.

Utilizing the triangle provides a much more straight-forward method for determining coefficients through binomial expansion rather than utilizing the binomial theorem method.

Given the impression of simplicity in constructing Pascal’s triangle, the number of uses and properties are astounding, due to the many properties and multiple patterns of numbers that are represented.

Performing calculations for the combinations and probabilities functions available through the use of Pascal’s triangle can be accomplished quickly and easily.

To illustrate: owners of the __TI 83/84 family of calculators__ can simply enter the first expression in the equation (such as 3 for the number of coin tosses in a probabilities example), hit the MATH key, then take the PRB (probabilities) function, and option 3 (nCr) to resolve the problem.

Other calculators such as the Casio scientific family have similar solutions for utilizing Pascal’s triangle for solutions.

To get more details on solving equations and binomial expansion with a calculator, there are a number of gracious individuals who have produced extremely __helpful videos__ on YouTube.

These offerings walk you step-by-step through entering the variables right down to the keys to press, with various types of problems provided for examples.

Pascal’s triangle provides many valuable patterns and applications for solving mathematical problems. Applying those principles through the use of sophisticated scientific/graphing calculators makes the process much easier and less prone to errors.

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