In the context of the binomial theorem, the letter k represents the index of summation and indicates the position of the term in the expansion of a binomial expression.
The binomial theorem states that for any positive integer n and any two terms a and b, the expansion of the expression (a + b)n can be written as:
(a + b)n = Σ (n choose k) * a(n-k) * bk where k ranges from 0 to n.
Here, n choose k (often denoted as C(n, k) or binomial coefficient) gives the number of ways to choose k elements from a set of n elements. It is calculated using the formula:
C(n, k) = n! / (k! * (n-k)!).
In each term of the expansion, k tells us how many times the term b is being multiplied out, while the term a is multiplied out (n-k) times. Therefore, varying k allows us to produce each term in the complete expansion, ranging from the first term with all a and no b (when k = 0) to the last term with no a and all b (when k = n).