To understand why the correct answer is option 'D', let's break down the problem step by step.

Given:
- The set contains 2n+1 elements.
- We want to find the number of subsets of this set that contain more than n elements.

Understanding subsets:
- A subset is a collection of elements from a given set.
- For example, if we have a set {1, 2, 3}, its subsets are: {}, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}.
- The empty set {} and the set itself {1, 2, 3} are also considered subsets.

Understanding the number of subsets:
- For a set with n elements, there are 2^n subsets.
- This is because each element has two choices: either it is included in a subset or it is not.

Approach:
- We have a set with 2n+1 elements, so the number of subsets would be 2^(2n+1).
- However, we need to find the number of subsets that contain more than n elements.

Explanation:
1. To find the number of subsets that contain more than n elements, we need to determine the number of subsets that have less than or equal to n elements.
2. The number of subsets with less than or equal to n elements is the sum of the subsets with 0, 1, 2, ..., n elements.
3. For a set with 2n+1 elements, the number of subsets with 0, 1, 2, ..., n elements is given by the binomial coefficient.
4. The binomial coefficient can be calculated using the formula C(n, r) = n! / (r! * (n-r)!), where n is the total number of elements and r is the number of elements in the subset.
5. So, the number of subsets with less than or equal to n elements would be C(2n+1, 0) + C(2n+1, 1) + C(2n+1, 2) + ... + C(2n+1, n).
6. Now, the total number of subsets with more than n elements can be calculated by subtracting the number of subsets with less than or equal to n elements from the total number of subsets (2^(2n+1)).
7. Therefore, the number of subsets with more than n elements would be 2^(2n+1) - (C(2n+1, 0) + C(2n+1, 1) + C(2n+1, 2) + ... + C(2n+1, n)).
8. Simplifying the expression further, we get 2^(2n+1) - (C(2n+1, 0) + C(2n+1, 1) + C(2n+1, 2) + ... + C(2n+1, n)) = 2^(2n+1) - (2^(2n+1) - C(2n+1, n+1)).
9. Finally, simplifying the expression, we get C(2n+1, n+1) = 2^(2n).
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