To find out how many subsets of the set [tex]\( B = \{1, 2, 3, 4\} \)[/tex] have exactly two elements, we need to use combinatorial principles.
The process involves finding combinations of 2 elements from the set [tex]\( B \)[/tex]. In combinatorial terms, this is often denoted as [tex]\( \binom{n}{k} \)[/tex], where [tex]\( n \)[/tex] is the total number of elements in the set, and [tex]\( k \)[/tex] is the number of elements we want to select. Here:
- [tex]\( n = 4 \)[/tex] (since the set [tex]\( B \)[/tex] has 4 elements)
- [tex]\( k = 2 \)[/tex] (since we want to form subsets containing exactly 2 elements)
The formula for combinations is:
[tex]\[
\binom{n}{k} = \frac{n!}{k!(n-k)!}
\][/tex]
Plugging in the values we have:
[tex]\[
\binom{4}{2} = \frac{4!}{2! \cdot (4-2)!} = \frac{4!}{2! \cdot 2!}
\][/tex]
First, we compute the factorials:
- [tex]\( 4! = 4 \times 3 \times 2 \times 1 = 24 \)[/tex]
- [tex]\( 2! = 2 \times 1 = 2 \)[/tex]
So,
[tex]\[
\binom{4}{2} = \frac{24}{2 \times 2} = \frac{24}{4} = 6
\][/tex]
Therefore, there are 6 subsets of the set [tex]\( B \)[/tex] that have exactly two elements. The subsets are:
[tex]\[
\{1, 2\}, \{1, 3\}, \{1, 4\}, \{2, 3\}, \{2, 4\}, \{3, 4\}
\][/tex]
So the correct answer is:
[tex]\[
6
\][/tex]
