How many ways can you arrange 8 people at a round table if clockwise and anticlockwise arrangements are considered different? Consider cases with no restrictions, with George and Harry sitting together, with Mary refusing to sit next to George, and with clockwise/anticlockwise arrangements considered equivalent.

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Question 6: How many ways can you arrange 8 people at a round table (clockwise and anticlockwise arrangements are considered different)

• a) With no restrictions For circular permutations where clockwise and anticlockwise arrangements are considered different, the number of ways to arrange $n$ distinct items is $(n-1)!$. Here, $n=8$. Step 1: Apply the formula for circular permutations. $(8-1)! = 7!$ Step 2: Calculate the factorial. $7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040$ There are $\boxed{5040 ways}$ to arrange 8 people with no restrictions.

• b) If George and Harry must sit next to each other Step 1: Treat George and Harry as a single unit. Now we are arranging $8-1=7$ units (6 individuals + the GH pair). Step 2: Arrange these 7 units around the table. The number of ways is $(7-1)! = 6!$. Step 3: Consider the internal arrangement of George and Harry within their unit. They can sit as (George, Harry) or (Harry, George), which is $2!$ ways. Step 4: Multiply the arrangements of the units by the internal arrangements of the pair. $6! \times 2! = (720) \times (2) = 1440$ There are $\boxed{1440 ways}$ if George and Harry must sit next to each other.

• c) If Mary refuses to sit next to George Step 1: Calculate the total number of arrangements with no restrictions (from part a), which is $7! = 5040$. Step 2: Calculate the number of arrangements where Mary does sit next to George. This is similar to part b. Treat Mary and George as a single unit. We are arranging $8-1=7$ units. This gives $(7-1)! = 6!$ ways. Within the unit, Mary and George can be arranged in $2!$ ways (MG or GM). So, arrangements where Mary does sit next to George = $6! \times 2! = 720 \times 2 = 1440$. Step 3: Subtract the arrangements where Mary sits next to George from the total arrangements. $5040 - 1440 = 3600$ There are $\boxed{3600 ways}$ if Mary refuses to sit next to George.

• d) If clockwise and anticlockwise arrangements are considered equivalent. For circular permutations where clockwise and anticlockwise arrangements are considered equivalent (symmetrical arrangements), the number of ways to arrange $n$ distinct items is $\frac{(n-1)!}{2}$. Here, $n=8$. Step 1: Apply the formula for symmetrical circular permutations. $\frac{(8-1)!}{2} = \frac{7!}{2}$ Step 2: Calculate the factorial and divide by 2. $\frac{5040}{2} = 2520$ There are $\boxed{2520 ways}$ if clockwise and anticlockwise arrangements are considered equivalent.