If `A=[1,2,3,4,5,6]` then how many subsets of A contain the element 2, 3 and 5?

Options

A. 4
B. 8
C. 16
D. 32

Answer

To find how many subsets of the set \( A = \{1, 2, 3, 4, 5, 6\} \) contain the elements 2, 3, and 5, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Required Elements**:
We need to ensure that the elements 2, 3, and 5 are included in every subset we consider.

2. **Create a New Set**:
Since 2, 3, and 5 must be included, we can think of them as fixed elements in our subsets. The remaining elements of set \( A \) are 1, 4, and 6. Thus, we can create a new set \( B = \{1, 4, 6\} \) which contains the elements that can either be included or excluded from the subset.

3. **Determine the Number of Subsets**:
Each of the elements in set \( B \) can either be included in a subset or not. Therefore, for each of the 3 elements in \( B \), there are 2 choices (include it or exclude it).

The total number of subsets that can be formed from set \( B \) is given by:
\[ 2^{n} \]
where \( n \) is the number of elements in \( B \). Here, \( n = 3 \).

So, the total number of subsets from set \( B \) is:
\[ 2^3 = 8 \]

4. **Combine with Fixed Elements**:
Each of these 8 subsets of \( B \) can be combined with the fixed elements 2, 3, and 5. Therefore, the total number of subsets of \( A \) that contain the elements 2, 3, and 5 is also 8.

### Final Answer:
The total number of subsets of \( A \) that contain the elements 2, 3, and 5 is **8**.