Six men and four women need to sit down for dinner at a circ

Six men and four women need to sit down for dinner at a circular table. In how many ways can this be done? In how many ways can this be done if no two women may sit next to each other? In how many ways can this be done if no two men may sit next to each other?

Solution

Six men and Four Women need to sit down for at a circular table

(a) In how many ways can this be done ?

In thise case there is no restriction , such as (all women are together, or no women are together etc)

Just we need how many ways these 10 people (6 men +4 women) sit at acircular table for dinner

this can be done (10 - 1) ! ways = 9 ! = 362880 ,

[Because the permutations of n person are n!, each permutation gives rise to n identical patterns for the table composition (n rotations of one chair at each step) so that you obtain the desired number: n!/n=n (n1)!/n=(n1)! ].

(b) In how many ways this can be done if no two women may sit next to each other ?

Firstly, 6 men can arrange in circular table in (6-1) ! ways = 5 ! = 120

Then, if we arrange each women between the men (because  no two women may sit next to each other) ,here 6 slots available between the men , it can done ⁶C₄  =15 ,

Also, these 4 women can arranged in 4! ways

Therefore, Total number of ways are 120 * 15 * 4! = 120*15 * 24 = 43200

(c) In how many ways this can be done if no two men may sit next to each other ?

In this case, firsly, arranege the 4 wen can sit (4-1)! ways =3 !

next, if we arrange each men between them, it not possible because there are only 4 slots are available for 6 men ( 4 men sit 4 places between two women, but remaing two men are left)

another case if we arrange 6 men in a circular table, then arrange each women between 2 men because no men next to each other, but here 3 men each next to each other,

Ans:It is Not possible, no two men may sit next to each other