Question

For all integers n ≤ 1, the ᵏ⁼⁰Σⁿ (nk) (-1)ᵏ = 0

In other words, for all of the subsets of {1, 2, ..., n }, the number of ones with even cardinality is the same as the number of odd ones.

Answer 1

The question is about the mathematical principle that for any set of integers, the subsets with even cardinality are equal in number to those with odd cardinality, relating to concepts from combinatorial math and the binomial theorem.

Step-by-step explanation:

The question revolves around a mathematical concept concerning the equality of the number of subsets with even cardinality to those with odd cardinality in a set of integers. Specifically, the question is about a series that sums up to zero, indicating an equal number of even and odd cardinality subsets for any integer n ≤ 1. This concept is closely related to the binomial theorem and combinatorial mathematics, as it deals with the counting of subsets.

The latter portion of the question's reference material discusses series expansions and the binomial theorem. The binomial theorem describes the expansion of a power of a binomial (a + b), which is relevant to understanding the properties of subsets with respect to evenness and oddness. The examples provided, like n² and concepts like cardinality of the subsets, are fundamental to grasping the deeper implications of the original question.