Let's consider the correct sigma notation for expanding [tex]\((-5m - 7n)^{19}\)[/tex]:
[tex]\[
(-5m - 7n)^{19} = \sum_{r=0}^{19} \binom{19}{r} (-5m)^r (-7n)^{19-r}
\][/tex]
To understand this step-by-step:
1. Binomial Expansion: The binomial expansion of [tex]\((a + b)^n\)[/tex] is given by:
[tex]\[
(a + b)^n = \sum_{r=0}^{n} \binom{n}{r} a^r b^{n-r}
\][/tex]
2. Identify Components:
- [tex]\(a = -5m\)[/tex]
- [tex]\(b = -7n\)[/tex]
- [tex]\(n = 19\)[/tex]
3. Substitute into the Formula: By substituting the identified components into the binomial expansion formula, we get:
[tex]\[
(-5m - 7n)^{19} = \sum_{r=0}^{19} \binom{19}{r} (-5m)^r (-7n)^{19-r}
\][/tex]
4. Expand: Now we expand this sum by identifying each term from [tex]\(r = 0\)[/tex] to [tex]\(r = 19\)[/tex]:
[tex]\[
\begin{align*}
&= \binom{19}{0} (-5m)^0 (-7n)^{19} + \binom{19}{1} (-5m)^1 (-7n)^{18} + \binom{19}{2} (-5m)^2 (-7n)^{17} + \cdots + \binom{19}{19} (-5m)^{19} (-7n)^0 \\
&= \binom{19}{0} (1) (-7n)^{19} + \binom{19}{1} (-5m) (-7n)^{18} + \binom{19}{2} (-5m)^2 (-7n)^{17} + \cdots + \binom{19}{19} (-5m)^{19} (1)
\end{align*}
\][/tex]
Taking this concept and matching it to your expressions, it is clear that the correct expansion and representation match the first expression:
[tex]\[
\sum_{r=0}^{19} \binom{19}{r} (-5m)^r (-7n)^{19-r}
\][/tex]
The remaining expressions are either syntactically incorrect or incorrectly apply the binomial theorem. For example:
- [tex]\(\sum_r^{19} \binom{19}{r} (-5m)^{19-\gamma} (-7n)^r\)[/tex]: incorrect because the notation uses [tex]\(\gamma\)[/tex] instead of [tex]\(r\)[/tex] consistently.
- [tex]\(\sum_{r=0}^{19} \binom{r}{19} (-5m)^{19-\gamma} (-7n)^r\)[/tex]: incorrect since the binomial coefficient is written incorrectly as [tex]\(\binom{r}{19}\)[/tex] rather than [tex]\(\binom{19}{r}\)[/tex].
Through the correct binomial expansion and substitution, we see that the right sigma notation representation for [tex]\((-5m - 7n)^{19}\)[/tex] is:
[tex]\[
\sum_{r=0}^{19} \binom{19}{r} (-5m)^r (-7n)^{19-r}
\][/tex]
