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CIRCULAR PERMUTATIONS

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

Remarks:

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.

Illustration 1:

Consider 23 different coloured beads in a necklace. In how many ways can the beads be placed in the necklace so that 3 specific beads always remain together?

Solution:

By theory, let us consider 3 beads as one. Hence we have, in effect, 21 beads,   ‘n’ = 21. The number of arrangements = (n-1)! = 20!

Also, the number of ways in which 3 beads can be arranged between themselves is 3! = 3 x 2 x 1 = 6.

Thus the total number of arrangements = (1/2). 20! 3!

Illustration 2:

In how many ways 10 boys and 5 girls can sit around a circular table so that no two girls sit together.