Question

The coefficient of $x^{10}$ in the expansion of $\left(x + \frac{2}{x} - 5 \right)^{12}$ is

• A. 1674
• B. 2132
• C. 1892
• D. 862

Hint:

Use multinomial theorem and track powers carefully; solve system of equations for exponents to find specific term.

Answer 1

The Correct Option is A

Expand $(x + \frac{2}{x} - 5)^{12}$ using multinomial theorem. Let the powers in the expansion be $x^a$, $(\frac{2}{x})^b$, $(-5)^c$ with $a + b + c = 12$. Power of $x$ in term is $a - b$. We want $a - b = 10$.
From $a + b + c = 12$ and $a - b = 10$, solve to find $a, b, c$.
Adding both equations: $2a + c = 22$. Since $a, b, c$ are non-negative integers with $a+b+c=12$, find integer solutions $(a,b,c)$ satisfying these constraints.
Possible $(a,b,c)$: $a=11, b=1, c=0$ and $a=10, b=0, c=2$ (other combos violate $a+b+c=12$).
Calculate coefficients for each using multinomial formula and sum them. This gives coefficient = 1674.