Question

The coefficient of \( x^{17} \) in

\[ (1 - x)^{13} (1 + x + x^2)^{12} \] is:

• A. \( ^{12}C_6 \)
• B. \( ^{9}C_7 \)
• C. 0
• D. 1
• E. \( ^{12}C_4 \)

Hint:

In problems like this, use the binomial and multinomial expansions and look for matching powers of \( x \) from both expressions. If no valid combination exists, the coefficient is zero.

Answer 1

The Correct Option is C

We are tasked with finding the coefficient of \( x^{17} \) in the expansion of \( (1 - x)^{13} (1 + x + x^2)^{12} \). To do this, we need to consider the expansions of both terms separately.
1. Expansion of \( (1 - x)^{13} \):
The binomial expansion of \( (1 - x)^{13} \) is given by: \[ (1 - x)^{13} = \sum_{k=0}^{13} \binom{13}{k} (-1)^k x^k \] Thus, the general term in this expansion is \( \binom{13}{k} (-1)^k x^k \). 
2. Expansion of \( (1 + x + x^2)^{12} \):
The expansion of \( (1 + x + x^2)^{12} \) can be found using the multinomial theorem. 
The general term in the expansion is: \[ \binom{12}{a,b,c} x^{a + b + 2c} \] where \( a, b, c \) are non-negative integers such that \( a + b + c = 12 \).
We need the product of the general terms from both expansions that will give \( x^{17} \). 
- The power of \( x \) from \( (1 - x)^{13} \) is \( k \), and the power of \( x \) from \( (1 + x + x^2)^{12} \) is \( a + b + 2c \).
- Therefore, we need to find \( k \) and \( a + b + 2c \) such that \( k + (a + b + 2c) = 17 \).
However, from the nature of the expansions, it is evident that no valid combination of terms will result in a power of \( x^{17} \), meaning the coefficient of \( x^{17} \) is 0.
Thus, the correct answer is option (C), 0.