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We are given the expression \((1 - 5x)^7\) and need to find the coefficient of \(x^3\) when the expression is expanded in ascending powers of \(x\).

### Step-by-Step Solution Using Binomial Expansion:

The binomial expansion of \((1 - 5x)^7\) is given by:

\[
(1 - 5x)^7 = \sum_{k=0}^{7} \binom{7}{k} (1)^{7-k} (-5x)^k
\]

This simplifies to:

\[
(1 - 5x)^7 = \sum_{k=0}^{7} \binom{7}{k} (-5)^k x^k
\]

### Finding the Coefficient of \(x^3\):
For \(x^3\), we need to look at the term where \(k = 3\).

The general term in the expansion is:

\[
T_k = \binom{7}{k} (-5)^k x^k
\]

Substituting \(k = 3\):

\[
T_3 = \binom{7}{3} (-5)^3 x^3
\]

First, calculate \(\binom{7}{3}\):

\[
\binom{7}{3} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35
\]

Now, compute \((-5)^3\):

\[
(-5)^3 = -125
\]

Therefore, the term for \(x^3\) is:

\[
T_3 = 35 \times (-125) x^3 = -4375 x^3
\]

Thus, the coefficient of \(x^3\) is **-4375**.

Would you like further details or have any questions?

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Here are 5 follow-up questions to deepen your understanding:

1. What is the general form of the binomial expansion?
2. How would the expansion change if the exponent was different, say \(n = 8\)?
3. What happens to the signs of the terms in the expansion if the binomial has a positive term instead, like \((1 + 5x)^7\)?
4. How can you find the coefficient of \(x^5\) in the same expansion?
5. What are the conditions for a binomial expansion to be valid?

**Tip:** Understanding binomial coefficients is key to mastering binomial expansions. The values can be found using combinations or Pascal's triangle.

When (1 - 5x)^7 is expanded in ascending powers of x, find the coefficient of x^3.

Learn how to find the coefficient of x^3 in the binomial expansion of (1 - 5x)^7. This problem uses the binomial expansion formula and combinatorics to solve for the coefficient, providing a step-by-step breakdown.

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