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The Binomial Formula and Binomial Coefficients

Factorial n

If n = 1, 2 , 3, ..... factorial n or n factorial is defined as
3.1            n! = 1 • 2• 3 • ..... • n
We also define zero factorial as
3.2             0! = 1

Binomial Formula for Positive Integral n

If n = 1, 2, 3, ..... then
3.3             (x + y)ⁿ = xⁿ + nx^{n - 1}y + [n(n - 1)/2!].x^{n - 2}y² + [n(n - 1)(n - 2)/3!].x^{n - 3}y³ + ..... + yⁿ
This is called the binomial formula. It can be extended to other values of n and then is an infinite series.

Binomial Coefficients

The result 3.3 can also be written
3.4             
where the coefficinets, called binomial coefficients, are given by
3.5             

Properties of Binomial Coefficients

3.6             
This leads to Pascal's triangle

3.7 - 3.15             

Multinomial Formula

3.16             
where the sum, denoted by ∑, is taken over all nonnegative integers n₁, n₂, ....., n_p for which n₁ + n₂ + ... + n_p = n.