Question

Determine and write the number of subsets in a set questions.make it little bit difficult. PROVIDE the answers also below

Answer 1

To determine the number of subsets in a set, we use the formula:

\[
\text{Number of subsets} = 2^n
\]

where \( n \) is the number of elements in the set.

Here are some questions that are a bit more challenging:

### Questions:

1. A set \( A \) contains 8 elements. How many subsets does set \( A \) have?

2. If set \( B \) consists of the letters in the word "MATH", how many subsets can be formed from set \( B \)?

3. Consider a set \( C \) where it has 6 elements and 2 of the elements are repeated. How many unique subsets can be created from set \( C \)?

4. A set \( D \) has 10 distinct integers. If we were to remove one integer from set \( D \), how many subsets would the new set have?

5. If a set \( E \) contains 12 elements, what would be the number of proper subsets of set \( E \)?

---

### Answers:

1. **For set \( A \)**:
\[
\text{Number of subsets} = 2^8 = 256
\]

2. **For set \( B \)** (letters in "MATH" are {M, A, T, H}, which has 4 distinct letters):
\[
\text{Number of subsets} = 2^4 = 16
\]

3. **For set \( C \)** (assuming it's {a, a, b, c, d, e} which has 5 unique elements {a, b, c, d, e}):
To calculate the number of unique subsets:
\[
\text{Number of unique subsets} = \frac{2^n}{k!} = \frac{2^5}{2!} = \frac{32}{2} = 16
\]
(where \( n \) is the total count of items including repetitions, and \( k \) is the count of the repeated item(s)).

4. **For set \( D \)** (initially has 10 distinct integers):
- Original number of subsets for 10 elements:
\[
2^{10} = 1024
\]
- After removing one integer, set \( D \) would have 9 elements:
\[
2^9 = 512
\]

5. **For set \( E \)** (containing 12 elements):
\[
\text{Total number of subsets} = 2^{12} = 4096
\]
To find proper subsets, we subtract the set itself:
\[
\text{Number of proper subsets} = 4096 - 1 = 4095
\]

Feel free to ask for more problems or any specific area you want to focus on!