Question

Find the coefficient of x^16
in the expression of (x^2− 2x)^10

Answer 1

Answer on Question #68659 - Math – Other

Question:

Find the coefficient of $x^{16}$ in the expression of $(x^2 - 2x)^{10}$.

Solution:

$$(x^2 - 2x)^{10} = \left(x(x - 2)\right)^{10} = x^{10}(x - 2)^{10} = x^{10} \sum_{k=0}^{10} \binom{10}{k} x^{10-k}(-2)^k$$

The coefficient of $x^{16}$ in the whole expression is the coefficient of $x^6$ in $\sum_{k=0}^{10} \binom{10}{k} x^{10-k}(-2)^k$, which is when $k = 4$:

$$c = \binom{10}{4} (-2)^4 = \frac{10!}{4!6!} (-2)^4 = \frac{10 \times 9 \times 8 \times 7}{1 \times 2 \times 3 \times 4} \times 16 = 210 \times 16 = 3360$$

Answer:

3360

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