Question

given that b = {1, 2, 3, 4}, how many subsets have exactly two elements?
○ 6
○ 5
○ 4

Explanation:

Step1: Recall combination formula

To find the number of subsets with exactly \( k \) elements from a set with \( n \) elements, we use the combination formula \( C(n,k)=\frac{n!}{k!(n - k)!} \), where \( n! = n\times(n - 1)\times\cdots\times1 \). Here, \( n = 4 \) (since set \( B \) has 4 elements) and \( k = 2 \) (we want subsets with exactly 2 elements).

Step2: Calculate factorials

First, calculate the factorials:

• \( 4! = 4\times3\times2\times1 = 24 \)
• \( 2! = 2\times1 = 2 \)
• \( (4 - 2)! = 2! = 2 \)

Step3: Substitute into combination formula

Substitute these values into the combination formula:
\[
C(4,2)=\frac{4!}{2!(4 - 2)!}=\frac{24}{2\times2}=\frac{24}{4} = 6
\]

Answer:

6 (corresponding to the option "6")