Binomial Theorem
Coefficient extraction via substitution; sum of coefficients
Grade Class 12
Question:
If $\left(x - 2 + \dfrac{1}{x}\right)^{30} = a_0 x^{30} + a_1 x^{29} + \ldots + a_{29}x + a_{30} x^{-1} + \ldots + a_{60}x^{-30}$ and $k = a_0 + a_1 + \ldots + a_{60}$. If $k - a_{30} = -{}^nC_r$, then $n + r$ is equal to
Step-by-Step Solution
Key Concept: Write $\left(x-2+\frac{1}{x}\right)^{30} = \frac{(x-1)^{60}}{x^{30}}$. Multiply both sides by $x^{30}$ to get $(x-1)^{60}$. Sum of coefficients: put $x=1$.
$(x-1)^{60}/x^{30}$: $k=0$, $a_{30}={}^{60}C_{30}$. $k-a_{30}=-{}^{60}C_{30}$, so $n=60$, $r=30$, $n+r=90$.
Correct Answer: 3