A class of problems that requires one to use permutations and combinations for computing probability has at its heart notion of sets and subsets. They are generally an abstract formulation of some concrete situation and require the application of counting techniques.
A is a set containing 10 elements. A subset `P_(1)` of A is chosen and the set A is chosen and the set A is reconstructed by replacing the elements of `P_(1)`. A subset `P_(2)` of A is chosen and again the set A is reconstructed by replacing the elements of `P_(2)`. This process is continued by choosing subsets `P_(1), P_(2), ... P_(10)`.
The number of ways of choosing subsets `P_(1), P_(2),...P(10)` is

Options

A. A) `4^(100)`
B. B) `2^(100)`
C. C) `2^(20)`
D. D) `4^(20)`

Answer

To solve the problem, we need to determine the number of ways to choose subsets \( P_1, P_2, \ldots, P_{10} \) from a set \( A \) that contains 10 elements.

### Step-by-Step Solution:

1. **Understanding the Set and Subsets**:
- The set \( A \) contains 10 elements. The total number of subsets of a set with \( n \) elements is given by \( 2^n \).
- Therefore, the number of subsets of set \( A \) is \( 2^{10} \).

**Hint**: Remember that each element can either be included or excluded from a subset, leading to \( 2^n \) possible combinations.

2. **Choosing Subsets**:
- For each subset \( P_i \) (where \( i = 1, 2, \ldots, 10 \)), we can choose any of the \( 2^{10} \) subsets of \( A \).
- This means that for \( P_1 \), there are \( 2^{10} \) choices.
- Similarly, for \( P_2 \), there are also \( 2^{10} \) choices, and this continues for each \( P_i \).

**Hint**: Each choice of subset is independent of the others, so you multiply the number of choices.

3. **Calculating Total Choices**:
- Since we are choosing 10 subsets \( P_1, P_2, \ldots, P_{10} \), and each has \( 2^{10} \) choices, the total number of ways to choose these subsets is:
\[ (2^{10})^{10} \]
- This simplifies to:
\[ 2^{10 \times 10} = 2^{100} \]

**Hint**: When you raise a power to a power, you multiply the exponents.

4. **Final Answer**:
- The total number of ways of choosing subsets \( P_1, P_2, \ldots, P_{10} \) is \( 2^{100} \).

### Conclusion:
Thus, the number of ways of choosing subsets \( P_1, P_2, \ldots, P_{10} \) is \( 2^{100} \).

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