Binomial coefficient

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The binomial coefficients arise when developing the successive terms in the binomial theorem

The number n is the power of the binomial theorem, the number r is the current number of the term in the outcome.

We write the binomial expansion and see

It contains a clear regularity. Each term has the general form

Coefficient × a^n−rb^r

in which

n = power of the binomium
r = number of the term

The expansion of the coefficients follows Pascal's triangle. A number is formed by the sum of the number there just above left and right

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

In each row the numbers increase, and then decrease in the same way. They can be calculated with the formula for combinations

in which

n = power of the binomium
r = number of the term