Home
Class 12
MATHS
(i) Find the coefficient of x^(3)y^(4)z^...

(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)`

Text Solution

Verified by Experts

(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)`
Promotional Banner

Topper's Solved these Questions

  • BINOMIAL THEOREM

    AAKASH INSTITUTE ENGLISH|Exercise Illustration|1 Videos

  • BINOMIAL THEOREM

    AAKASH INSTITUTE ENGLISH|Exercise Try Yourself|20 Videos

  • APPLICATION OF INTEGRALS

    AAKASH INSTITUTE ENGLISH|Exercise Assignment Section - I Aakash Challengers Questions|2 Videos

  • COMPLEX NUMBERS AND QUADRATIC EQUATIONS

    AAKASH INSTITUTE ENGLISH|Exercise section-J (Aakash Challengers Qestions)|13 Videos

Similar Questions

Explore conceptually related problems

In the expansion of (x +y + z)^25

Find the coefficient of x^3y^4z^5 in the expansion of (xy+yz+zx)^6

Cofficient of x^(3)y^(10)z^(5) in expansion of (xy+yz+zx)^(6) is

The coefficient of x^2y^4z^2 in the expansion of (2x - 3y + 4z)^9 is

The coefficient of x^2 y^5 z^3 in the expansion of (2x + y + 3z)^10 is

The coefficient of x^(8) y^(6) z^(4) in the expansion of (x + y + z)^(18) , is not equal to

Find the coffiecient of x^2 y^3 z^4 w in the expansion of (x-y-z+w)^(10)

Statement-1: The coefficient of a^(3)b^(4)c^(3) in the expansion of (a-b+c)^(10) is (10!)/(3!4!3!) Statement-2: The coefficient of x^(p)y^(q)z^(r) in the expansion of (x+y+z)^(n) is (n!)/(p!q!r!) for all integer n.

In the expansion of (3x+2y-z)^(8) , the coefficients of x^(2)y^(3)z^(3) is

Find the coefficient of x^2 y^3 z^4 in the expansion of (ax -by +cz)^9 (b) a^2 b^3 c^4 d in the expansion of (a -b-c +d)^10