In Example 28, the number of ways of choosing A
and B such that B is a subset of A, is

Options

A. `2^(n)`
B. `3^(n)`
C. `""^(2n)C_(n)`
D. none of these

Explanation

To solve the problem of finding the number of ways of choosing sets A and B such that B is a subset of A, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Elements**: Let the set A have n elements. We denote the number of elements in set A as n.

2. **Choosing R Elements for A**: We can choose R elements from the n elements to form the set A. The number of ways to choose R elements from n is given by the binomial coefficient \( \binom{n}{R} \).

3. **Choosing B as a Subset of A**: Once we have chosen R elements for set A, we need to choose set B such that B is a subset of A. The number of subsets of a set with R elements is \( 2^R \) (including the empty set).

4. **Total Combinations for Each R**: For each specific choice of R, the total number of ways to choose A and B is given by:
\[ \text{Total ways for a specific R} = \binom{n}{R} \cdot 2^R \]

5. **Summing Over All Possible R**: Since R can vary from 0 to n, we need to sum the total combinations for all possible R:
\[ \text{Total ways} = \sum_{R=0}^{n} \binom{n}{R} \cdot 2^R \]

6. **Recognizing the Binomial Theorem**: The expression \( \sum_{R=0}^{n} \binom{n}{R} \cdot 2^R \) can be recognized as the expansion of \( (1 + 2)^n \) according to the binomial theorem.

7. **Final Calculation**: Therefore, we have:
\[ \sum_{R=0}^{n} \binom{n}{R} \cdot 2^R = 3^n \]

Thus, the number of ways of choosing sets A and B such that B is a subset of A is \( 3^n \).

### Final Answer:
The number of ways of choosing A and B such that B is a subset of A is \( 3^n \).

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