Circular Permutation 11th Class Maharashtra Board
For example,consider a n= 4
Thus given a single circular arrangement of a four object it considered to four different linear arrangements in a row Let the circular arrangement be m. we know that the total number of linear arrangement of four different object is 4! Circular Permutation 11th Class Maharashtra Board
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| Circular Permutation 11th Class Maharashtra Board |
Raleated Post :- Permutation and combination
Each arrangement in row corresponding to some circular arrangement
Therefore number of circular arrangement is
`m=\frac{\text{4!}}{\text{4}}=3!`
Similarly, each circular arrangement of n objects corresponds to n different arrangements in a row.
Therefore number of circular arrangements is
`\frac{\text{n!}}{n}=(n-1)!`
Note -
1) a circular arrangement does not have a fixed starting point it and any rotation of it is considered to be the same circular arrangement this arrangement will be the same with respect to each other if rotation is made but clockwise and anticlockwise arrangement at the front second point
2) if clockwise and anticlockwise circular arrangement are considered to be the same then a circular arrangement correspond to two and different linear arrangement does the number of circular arrangement is
`\frac{\text{n!}}{\text{2n}}=\frac{\text{(n-1)!}}{\text{2}}`
Theorem:
The number of circular arrangement soft and objects of which objects are alike identical, is given by and (n-1)!/m!
REMARK:
The number of of circular permutations of our objects taken from n distinct objects can be found under two different conditions as follows.
A)When clockwise and anticlockwise arrangement are considered to be different than the required number of circular arrangements is given by
$\frac{^nP_r}{r}$
B)when clockwise and anticlockwise arrangements are not to be considered different, then the required number of circular arrangements is given by
$\frac{^nP_r}{2r}$
Example: In how many ways can 8 students be arranged at a round table so that 2 particular students are together if
(i) students are arranged with respect to each other?
(ii)students are arranged with respect to the table?
Solution:
Considering those 2 particular students as one students we have 7 students.
(i)In circular arrangement 7 students can be arranged at a round table, in 6! ways and 2 students can be arranged among themselves in $^2P_2$=2! ways.
Hence, the required number of ways in which two particular students come together= 6! * 2!= 1440
(ii)Here the arrangement is like an arrangement in a row.
so, 7 students can be arranged in 7! ways and 2 students can be arranged in 7! ways and 2 students can be arranged amongst themselves in $^2P_2$=2! ways.
Hence, the required number of ways in which two particular students come together
$= 7! \times 2!$
= 10080.
EXAMPLE 2: In how many ways 6 women can be seated at a round table so that every man has woman by his side.
SOLUTION:
3 women have (3-1)!= 2 ways of circular seatings.
In each seating a women has one place on each side for a man. Thus there are 6 place on each side for a man. Thus there are 6 different places for 6 men which can be filled in 6!*2= 720*2=1440
EXAMPLE 3: Find the number of ways in which 12 different flowers can be arranged in a garland so that four different flowers can be arranged in a garland so that 4 particular flowers are always together.
ANSWER:
Considering 4 particular flowers as a single flower, we have 9 flowers which can be arranged to form a garland in 8! ways. But 4 particular flowers can be arranged in 4! ways. Hence the required number of ways =$\frac{1}{2!}$$(8!\times2!)=48340$
