if sum of the coeficients in the expansion of `(2x+3y-2z)^(n)` is 2187 then the greatest coefficient in the expansion of `(1+x)^(n)`

Options

A. 30
B. 40
C. 28
D. 35

Answer

To solve the problem, we need to follow these steps:

### Step 1: Understand the sum of coefficients in the expansion
The sum of the coefficients in the expansion of a polynomial \( (a + b + c)^n \) can be found by substituting \( x = 1 \) for each variable in the polynomial.

For the expression \( (2x + 3y - 2z)^n \), the sum of the coefficients is given by:
\[ 2(1) + 3(1) - 2(1) = 2 + 3 - 2 = 3 \]
Thus, the sum of the coefficients in the expansion is \( 3^n \).

### Step 2: Set up the equation
We know from the problem that the sum of the coefficients equals 2187:
\[ 3^n = 2187 \]

### Step 3: Solve for \( n \)
To find \( n \), we need to express 2187 as a power of 3:
\[ 2187 = 3^7 \]
Thus, we have:
\[ 3^n = 3^7 \implies n = 7 \]

### Step 4: Find the greatest coefficient in the expansion of \( (1 + x)^n \)
The greatest coefficient in the expansion of \( (1 + x)^n \) is given by the binomial coefficient \( \binom{n}{k} \) where \( k \) is the integer closest to \( \frac{n}{2} \).

For \( n = 7 \):
\[ k = \left\lfloor \frac{7}{2} \right\rfloor = 3 \quad \text{or} \quad k = \left\lceil \frac{7}{2} \right\rceil = 4 \]

### Step 5: Calculate the coefficients
Now we calculate the coefficients:
\[ \binom{7}{3} = \frac{7!}{3!(7-3)!} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35 \]
\[ \binom{7}{4} = \frac{7!}{4!(7-4)!} = \frac{7 \times 6 \times 5 \times 4}{4 \times 3 \times 2 \times 1} = 35 \]

### Step 6: Conclusion
The greatest coefficient in the expansion of \( (1 + x)^7 \) is \( 35 \).

### Final Answer
The greatest coefficient in the expansion of \( (1 + x)^n \) is \( 35 \).

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