We can solve this problem using the definition of multinomial theorem
Multinomial Theorem For a Positive Integral Index
If
x1,x2 , ... ,
xk are real numbers, then fro all n ∊ N
(x1+x2+...+xk)n =
Σr1+r2+..+rk =
nx1r1.x2r2...xkrk where
r1,r2 , ... ,
rk are all non - negative integers
Some useful results on multinominal theorem
sum of all the coefficients is obtained by putting all the variables
xi equal to 1 and it is equal to
nm By using this result we can find the greatest coefficient in the expansion of
(1+x)n We have
the sum of the coefficients in the expansion of
(x−2y+3z)n is 128
We have,
(x−2y+3x)n Sum of the coefficients in the expansion of (putting x and y as 1)
(1−2+3)n = 128
i.e.,
2n =
27 (since, 128 =
27)
⇒ n = 7
Therefore, the greatest coefficient in
(1+x)7 is
C3 =
= 7 × 5 = 35
Hence
If the sum of the coefficients in the expansion of
(x−2y+3z)n is 128, then the greatest coefficient in the expansion of
(1+x)n is 35.