Pascal 's Triangle : Special Mathematical Properties

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Conclusions: As expected, my hypothesis was correct: If Pascal’s triangle has special mathematical properties (relationship with Binomial Theorem, the sum of the numbers in any row of Pascal’s triangle is a power of two, and the number below two entries across from one another is equal to the sum of both numbers in Pascal’s triangle) , then we can demonstrate, possibly even prove, these properties/relationships through examples, algebraic proofs, and combinatorial proofs because each property concerns binomial expansion, combinations, and combinatorics, thus we can use formulas, counting methods, and simple algebra to prove and illustrate such properties. Pascal’s triangle limits the amount of work necessary in expanding a binomial expression with the Binomial Theorem by eradicating the strenuous process of using the combination formula (n Choose k = n!/k!(n - k)!) to find the binomial coefficients. Effortlessly replace the combinations (n Choose k) with the elements/numbers of the row of Pascal’s triangle, n being the row number, to get the coefficients. Simply put, look at Pascal’s triangle. However, many argue that the construction of Pascal’s triangle for large values of n is a tedious process, but the use of the powers of eleven property contradicts this notion. This property states that: if a row is made into a single number by using each element as a digit of the number (carrying over when an element itself has more than one digit), the number is equal to 11 to the
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