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Pascal's Triangle and nCr
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Pascal's triangle and binomial coefficients, using factorials and \(^{n}C_{r}=\frac{n!}{r!(n-r)!}\) to find combinations and the coefficients in binomial expansions.
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Factorials
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Factorials are shortcut used to express decreasing products of integers such as
5⋅4⋅3⋅2⋅1=5!
The definition is
n!=n×(n−1)×⋯×2×1🚫
Fractions of factorials as partial products
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k!n!=n×(n−1)×⋯×(k+2)×(k+1)
Binomial Coefficient nCr
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The number of ways to choose r items from a set of n items can be expressed as (nr), nCr or nCr. The number of combinations can be calculated using the formula
nCr=r!(n−r)!n!📖
Pascal's Triangle
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Pascal's triangle is a triangular array where each number is the sum of the two directly above it, beautifully revealing the coefficients of binomial expansions. Each term corresponds to a specific value of nCr. Its symmetry and simple construction make it a powerful tool for exploring combinatorial relationships and probability.
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