Practice Questions
Test your understanding of arrangements and necklaces with these circular-focused questions.
In how many ways can a president and 4 directors be seated around a circular table?
Total members = $1 + 4 = 5$. Circular permutations for $n$ distinct objects = $(n-1)! = 4! = 24$.
Find the number of ways in which 10 different flowers can be arranged to form a garland.
For a garland, clockwise and anti-clockwise are the same. Result: $(10-1)! / 2 = 362,880 / 2 = 181,440$.
In how many ways can 6 people be seated around a table such that two particular people always sit together?
Treat the 2 people as 1 unit. Total units = $4 + 1 = 5$. Arrange in a circle: $4! = 24$. But the 2 can swap internally (2 ways). Result: $24 \times 2 = 48$.
How many ways can 5 boys and 5 girls be seated around a circle so that no two girls are together?
First seat the 5 boys in $(5-1)! = 4! = 24$ ways. There are 5 gaps between the boys. The 5 girls can be seated in these 5 gaps in $5! = 120$ ways. Result: $24 \times 120 = 2,880$.
In how many ways can 8 people sit around a round table?
$(8-1)! = 7! = 5040$ ways.
In how many ways can 4 couples sit around a circular table if each couple sits together?
Treat each couple as 1 unit. 4 units in a circle: $(4-1)! = 6$. Each couple can swap internally in 2 ways: $2^4 = 16$. Total = $6 \times 16 = 96$.
In how many ways can 7 people sit around a table if two particular persons must not sit adjacent?
Total arrangements = $6! = 720$. Arrangements where 2 sit together = $5! \times 2 = 240$. Not adjacent = $720 - 240 = 480$.
In how many ways can 6 people sit in a circle if clockwise and anticlockwise are considered the same?
$(6-1)! / 2 = 5! / 2 = 120 / 2 = 60$.
In how many ways can 5 men and 3 women sit around a table such that no two women sit together?
Arrange 5 men in $(5-1)! = 24$ ways. 5 gaps available, choose 3 for women: $5P_3 = 60$. Total = $24 \times 60 = 1440$.
How many ways can 6 beads of different colours be arranged to form a necklace?
Necklace = $(6-1)! / 2 = 120 / 2 = 60$.
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