A family consist of grandfather 6 son and 4 grand children . t5o be seated in a row for a dinner . the grand chidren wish to occupy the two seats at each end and the grandfater refuse to have a grandchildren on either side of him. determine the no of ways in which they can be seated for dinner.

possible ways to sit totle 10 members sit for dinner in a row is 10! =3628800

but,

The grand chidren wish to occupy the two seats at each end and the grandfater refuse to have a grandchildren on either side of him,

if the grandchildren sit either side of him ,2!*8!(1unit+6members+1unit)*2!=161280

Finally the number of ways they can be seated for dinner are  =3628800 - 161280 =3467520

hi utkarsh

the toatal number of seats in the row is 11

let us label as L1 , L2 , L3, ......L 11

in L1 & L11 the position will be ocupied by 2 of 4 grandchildren

so the total no. of ways are P ( 4,2)

group Si G.F Sj  together     S1 ,.... S6  are sons  G.F is grandfather i , j are generalised terms

as 1 and arrange it in 7 places we can do it by

P(7,1) and 2 sons on the either side can be arranged in P(6,2)

now remaining 6 places can be filled in any way by remaining 2 G.C ''S and 4 sons

so no. of ways are 6!

now applying the product rule we get

P(4,2)*P(7,1)*P(6,2)*6!