answer all questions with detailed explanation

You are given two boxes R and B. Box R contains n red balls (say numbered from 1 to n) and box B contains n blue balls (say numbered from 1 to n). In each step, one random ball is chosen from box R and one random ball is chosen from box B and the chosen balls are placed in opposite boxes (i.e., the ball chosen from R is placed in B and the ball chosen from B is placed in R).

1. What is the expected number of red balls in box R after k steps. (Hint: There are two approaches to solve this problem. One is to use linearity of expectation and other is using conditional expectation.)

2. What is the expected number of red balls in box R as the number of steps tends to infinity.