Find the coefficient of $x^{3}$ in the expansion of $(2 x - 5)^{6}$ .

medium 2 marks Class 11 · Mathematics · Binomial Theorem

Answer

-20000

The correct answer is -20000 — Compute $(3 6) = 20$ , $2^{3} = 8$ , $(- 5)^{3} = - 125$ and multiply: $20 \times 8 \times (- 125) = - 20000$ .

Step-by-step solution

Write the expansion: $(2 x - 5)^{6} = \sum_{k = 0}^{6} (k 6) (2 x)^{, 6 - k} (- 5)^{, k}$ .
The power of $x$ in the $(k^{t h})$ term is $(6 - k)$ . For it to be $3$ , set $6 - k = 3$ , giving $k = 3$ .
Substitute $k = 3$ : coefficient $= (3 6), 2^{3}, (- 5)^{3}$ .
Compute $(3 6) = 20$ , $2^{3} = 8$ , $(- 5)^{3} = - 125$ and multiply: $20 \times 8 \times (- 125) = - 20000$ .

Common mistake: Forgetting the negative sign of

(- 5)^{3}

Concept tested: Coefficient extraction from binomial expansion