The binomial theorem for a positive integer index n reads: (x + y)^n = sum_{k = 0}^n {n choose k} x^{n-k} y^k, where {n choose k} is the binomial coefficient, d...
The binomial theorem for a positive integer index n reads:
(x + y)^n = sum_{k = 0}^n {n choose k} x^{n-k} y^k,
where {n choose k} is the binomial coefficient, defined as:
{n choose k} = n! / (k! * (n-k)!).
Using this formula, we can expand (x + y)^n into a sum of powers of x and y. Each term in the sum corresponds to a specific choice of k elements from the set {1, 2, 3, ..., n}.
For instance, when n = 2, the binomial theorem expands to:
(x + y)^2 = x^2 + 2xy + y^2.
As we can see from this example, the binomial theorem allows us to express (x + y)^n as a weighted sum of powers of x and y. The weight of each term is determined by the binomial coefficient {n choose k}.
Using the binomial theorem, we can find the values of (x + y)^n for various values of n and x and y