Answered

Get the information you need with the help of IDNLearn.com's expert community. Ask anything and get well-informed, reliable answers from our knowledgeable community members.

Find the number of ways in which 5 people A, B, C, D, and E can be seated at a round table, such that:

a. A and B always sit together.
b. C and D never sit together.

Sagot :

GinnyAnsweridn
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].
We value your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. IDNLearn.com has the solutions you’re looking for. Thanks for visiting, and see you next time for more reliable information.