Step 1: Write down the general expansion
We need to find the coefficient of xn in the product:
E=(1+x)(1−x)n
First, let's expand (1−x)n using the Binomial Theorem:
(1−x)n=nC0−nC1x+nC2x2−⋯+(−1)n−1nCn−1xn−1+(−1)nnCnxn
Step 2: Multiply by (1+x)
Now, substitute this expansion back into the main expression:
E=(1+x)[nC0−nC1x+⋯+(−1)n−1nCn−1xn−1+(−1)nnCnxn]
To find the total coefficient of xn, we look at the two parts of the multiplication:
When 1 multiplies the xn term:
1×[(−1)nnCnxn]⟹Coefficient=(−1)nnCn
When x multiplies the xn−1 term:
x×[(−1)n−1nCn−1xn−1]⟹Coefficient=(−1)n−1nCn−1
Step 3: Combine and simplify the coefficients
Total coefficient of xn=(−1)nnCn+(−1)n−1nCn−1
We know standard combinatorial values:
Substitute these values into the equation:
Coefficient of xn=(−1)n(1)+(−1)n−1(n)
To make it match the options, let's factor out (−1)n. Remember that (−1)n−1=−1(−1)n=−(−1)n:
Coefficient of xn=(−1)n(1)−(−1)n(n)
Coefficient of xn=(−1)n(1−n)
Conclusion
The coefficient of xn is (−1)n(1−n).
Correct Option: (b) (−1)n(1−n)
