Binomial Theorem

Expands a binomial expression raised to any positive integer power n.

Gives the total number of terms in the expansion of (a + b)^n.

Yields the (r + 1)-th term in the expansion of (a + b)^n.

Specifies the middle term(s) of the expansion based on whether n is even or odd.

Identifies the index with the maximum binomial coefficient for a given power n.

States that coefficients equidistant from the beginning and end of the expansion are equal.

Gives the ratio between two consecutive binomial coefficients of index n.

Absorbs the multiplier r into the combination by decrementing both indices.

Rewrites division of a combination by r + 1 as an incremented combination divided by n + 1.

Computes the total sum of all binomial coefficients for a power n.

States that the sum of even-indexed coefficients equals that of odd-indexed coefficients, both equal to 2^(n-1).

Evaluates the convolution sum of binomial coefficients across two independent powers.

Gives the general term in the multinomial expansion of (x_1 + x_2 + ... + x_k)^n.