Circular permutations
circular permutations formula & Applications
There are arrangements in closed loops also, called as circular arrangements.
Suppose n persons (a1, a2, a3,…,an) are to be arranged around a circular table. The total number of circular arrangements of n persons is 
Distinction between clockwise and anti-clockwise Arrangements:
Consider the following circular arrangements:
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In figure I, the order is clockwise whereas in figure II, the order is anti-clock wise. These are two different arrangements. When distinction is made between the clockwise and the anti-clockwise arrangements of n different objects around a circle, then the number of arrangements = (n – 1)! |
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But if no distinction is made between the clockwise and the anti-clockwise arrangements of n different objects around a circle, then the number of arrangements is 
For an example, consider the arrangements of beads (all different) on a necklace as shown in figures A and B.
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Look at (A) having 3 beads x1, x2, x3 as shown. Flip (A) over on its right. We get (B) at once. However, (A) and (B) are really the outcomes of one arrangement but are counted as two different arrangements in our calculation. To nullify this redundancy, the actual number of different arrangements is (n-1)!/2. |
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When the positions are numbered, circular arrangement is treated as a linear arrangement.
In a linear arrangement, it does not make difference whether the positions are numbered or not.
Example : In how many ways 10 boys and 5 girls can sit around a circular table so that no two girls sit together.
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Detail Explanation :10 boys can be seated in a circle in 9! ways. There are 10 spaces inbetween the boys, which can be occupied by 5 girls in 10p5 ways. Hence total number of ways
= 9! 10p5 = |
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