(i) Find the coefficient of `x^(3)y^(4)z^(2)t^(5)` in the expansion of `(x-y+z-t)^(14)`.
(ii) Find the coefficient of `x^(10)y^(12)z^(8)` in the expansion of `(xy+yz+zx)^(15)`

Explanation

(i) We observe that 3+4+2+5=14. hence the coefficient of `x^(3)y^(4)z^(5)t^(5)` in the expansion of `(x-y+z-t)^(14)=(14!)/(3!4!2!5!)(-1)^(4)(-1)^(5)` [`because` coefficient of y and t are -ve]
`=(14!)/(5!4!3!2!)`
(ii) we have 10+12+8=30 and `(xy+yz+zx)^(15)=sum(15!)/(n_(1)!n_(2)!n_(3)!)(xy)^(n_(1))(yz)^(n_(2))(zx)^(n_(3))`
where `n_(1),n_(2),n_(3)` are non-negative integers such that `n_(1)+n_(2)+n_(3)=15`
`=sum(15!)/(n_(1)!n_(2)!n_(3)!)x^(n_(1)+n_(3))y^(n_(1)+n_(2))z^(n_(2)+n_(3))`
For the coefficient of `x^(10)y^(12)z^(8)`, we must have `n_(1)+n_(3)=10,n_(1)+n_(2)=12,n_(2)+n_(3)=8`
hence `n_(1)=7,n_(2)=5&n_(3)=3`
Consequently the requried coefficient of `x^(10)y^(12)z^(8)` in `(xy+yz+zx)^(15)`
`(15!)/(7!5!3!)=.^(15)P_(5)`