1 Combinatorics Intro: finite sets, characteristic vectors, permutations, cycles

Lecture 1 Combinatorics Introduction: finite sets, subsets, characteristic vectors, permutations, disjoint cycles decomposition

[sleepingforever]

While proving the cardinality of power set, every step was very rigorous but at the last step just intuition was used to get the cardinality as (2^n). Is there a way to do that step rigorously as well?