Answered

Get the answers you need at Westonci.ca, where our expert community is dedicated to providing you with accurate information. Experience the ease of finding accurate answers to your questions from a knowledgeable community of professionals. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.

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 :

GinnyAnswer
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].
Thanks for stopping by. We are committed to providing the best answers for all your questions. See you again soon. We appreciate your time. Please come back anytime for the latest information and answers to your questions. We're dedicated to helping you find the answers you need at Westonci.ca. Don't hesitate to return for more.