UPSEE Solved Model Paper 6

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 = n

n!

r1!r2!...rk!

x1r1.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

7

‌

C3 =

7×6×5

3×2×1

= 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.