In , Example 28 , the number of ways of choosing A
and B such that A and B have equal number of elements, is

Options

A. `2^(n)`
B. `3^(n)`
C. `(2^(n))^(2)`
D. `""^(2n)C_(n)`

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 and both have an equal number of elements, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**: We need to choose two sets, A and B, such that B is a subset of A. The number of elements in A is denoted by \( r \), where \( 0 \leq r \leq n \).

2. **Choosing Set A**: The number of ways to choose set A from a total of n elements is given by the binomial coefficient \( \binom{n}{r} \). This represents the number of ways to select r elements from n.

\[ \text{Ways to choose A} = \binom{n}{r} \]

3. **Choosing Set B**: Once we have chosen set A with r elements, we can choose set B from these r elements. The number of subsets of a set with r elements is \( 2^r \), which includes the empty set and all combinations of the r elements.

\[ \text{Ways to choose B} = 2^r \]

4. **Total Ways for Fixed r**: The total number of ways to choose sets A and B for a fixed r is the product of the ways to choose A and B.

\[ \text{Total ways for fixed r} = \binom{n}{r} \cdot 2^r \]

5. **Summing Over All Possible r**: Since r can vary from 0 to n, we need to sum the total ways for all possible values of r.

\[ \text{Total ways} = \sum_{r=0}^{n} \binom{n}{r} \cdot 2^r \]

6. **Using the Binomial Theorem**: The summation \( \sum_{r=0}^{n} \binom{n}{r} \cdot 2^r \) can be simplified using the Binomial Theorem, which states that:

\[ (x + y)^n = \sum_{r=0}^{n} \binom{n}{r} x^r y^{n-r} \]

By setting \( x = 2 \) and \( y = 1 \), we get:

\[ (2 + 1)^n = 3^n \]

Therefore,

\[ \sum_{r=0}^{n} \binom{n}{r} \cdot 2^r = 3^n \]

7. **Final Result**: Thus, the total number of ways of choosing sets A and B such that B is a subset of A is:

\[ \text{Total ways} = 3^n \]

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

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