Sure, let's solve this problem step-by-step.
Part (a): A and B always sit together
To solve the part where A and B always sit together, we can treat A and B as a single unit. Let's call this combined unit "AB."
1. When A and B are treated as a single unit, we now have four units to arrange: (AB), C, D, and E.
2. In a round table, the number of ways to arrange [tex]\( n \)[/tex] distinct objects is [tex]\((n-1)!\)[/tex] because the arrangement is circular and rotations of the circle are considered identical.
3. Thus, we have 4 units to arrange around the table, which can be done in [tex]\((4-1)!\)[/tex] or [tex]\(3!\)[/tex] ways.
4. Within the unit "AB", A and B can switch places. So, there are 2 ways to arrange A and B within the combined unit.
Therefore, the total number of ways in which A and B always sit together is:
[tex]\[ 3! \times 2 = 6 \times 2 = 12 \][/tex]
Part (b): C and D never sit together
To find the number of ways where C and D never sit together:
1. First, we calculate the total number of ways to arrange 5 people at a round table. This is given by [tex]\((5-1)!\)[/tex] or [tex]\(4!\)[/tex] ways:
[tex]\[ 4! = 24 \][/tex]
2. Next, we find the number of ways in which C and D sit together. Similar to part (a), we treat the pair (CD) as a single unit.
- We now have 4 units to arrange: (CD), A, B, and E. These can be arranged in [tex]\((4-1)!\)[/tex] or [tex]\(3!\)[/tex] ways.
- Within the unit "CD", C and D can switch places. Thus, there are 2 ways to arrange C and D within the combined unit.
3. Therefore, the number of ways in which C and D sit together is:
[tex]\[ 3! \times 2 = 6 \times 2 = 12 \][/tex]
4. To find the number of ways where C and D never sit together, subtract the number of ways where C and D sit together from the total arrangements:
[tex]\[ 24 - 12 = 12 \][/tex]
So, the total number of ways in which C and D never sit together is:
[tex]\[ 12 \][/tex]
In summary:
- The number of ways in which A and B always sit together is [tex]\( 12 \)[/tex].
- The number of ways in which C and D never sit together is [tex]\( 12 \)[/tex].
