2

I have an expression of the sum of the binomial coefficients in the polynomial expansion of (x - y)^{4}. (1 is substituted for x and y.) The equivalent expressions for this sum is to be written on several lines. There is the term

+ \left[(-1)^{3} \binom{3}{2} + (-1)^{3} \binom{3}{3}\right] + (-1)^{4}\binom{3}{3}

in the second line. I want the expression in the second line separated into two lines, and I want the + in

+ \left[(-1)^{1} \binom{3}{0} + (-1)^{1} \binom{3}{1}\right]

to be aligned vertically with the + in the previous expression. (I would like a similar separation in the following line.)

\documentclass{amsart}
\usepackage{amsmath}



\setlength{\oddsidemargin}{0.0in}
\setlength{\evensidemargin}{0.0in} \setlength{\textwidth}{6.1in}
\setlength{\topmargin}{0.0in} \setlength{\textheight}{9in}



\begin{document}


\begin{align*}
\sum_{i=0}^{4} (-1)^{i} \binom{4}{i}
&= (-1)^{0} \binom{4}{0} + (-1)^{1} \binom{4}{1} + (-1)^{2} \binom{4}{2} + (-1)^{3} \binom{4}{3} + (-1)^{4} \binom{4}{4} \\
&\begin{alignedat}{2}
=(-1)^{0} \binom{3}{0} + \left[(-1)^{1} \binom{3}{0} + (-1)^{1} \binom{3}{1}\right]
+ \left[(-1)^{2} \binom{3}{1} + (-1)^{2} \binom{3}{2}\right] \\
&+ \left[(-1)^{3} \binom{3}{2} + (-1)^{3} \binom{3}{3}\right] + (-1)^{4}\binom{3}{3} \\
= \left[(-1)^{0} \binom{3}{0} + (-1)^{1} \binom{3}{0}\right] + \left[(-1)^{1} \binom{3}{1} + (-1)^{2}\binom{3}{1}\right]
+ \left[(-1)^{2} \binom{3}{2} + (-1)^{3}\binom{3}{2}\right] + \left[(-1)^{3} \binom{3}{3} + (-1)^{4}\binom{3}{3}\right] \\
&= 0. \ \rule{1.5ex}{1.5ex}
\end{alignedat}
\end{align*}

\end{document}
3

I suggest you use a couple of aligned environments inside the align* environment.

enter image description here

\documentclass{amsart}
\usepackage{amssymb} % for "\blacksquare" macro

\setlength{\oddsidemargin}{0.0in}
\setlength{\evensidemargin}{0.0in} 
\setlength{\textwidth}{6.1in}
\setlength{\topmargin}{0.0in} 
\setlength{\textheight}{9in}

\begin{document}

\begin{align*}
\sum_{i=0}^{4} (-1)^{i} \binom{4}{i}
&= (-1)^{0} \binom{4}{0} + (-1)^{1} \binom{4}{1} + (-1)^{2} \binom{4}{2} + (-1)^{3} \binom{4}{3} + (-1)^{4} \binom{4}{4} \\
&=(-1)^{0} \binom{3}{0} 
\!\begin{aligned}[t] 
&+ \left[(-1)^{1} \binom{3}{0} + (-1)^{1} \binom{3}{1}\right]
+ \left[(-1)^{2} \binom{3}{1} + (-1)^{2} \binom{3}{2}\right] \\
&+ \left[(-1)^{3} \binom{3}{2} + (-1)^{3} \binom{3}{3}\right] + (-1)^{4}\binom{3}{3} 
\end{aligned}\\
&= \!\begin{aligned}[t]
  &\left[(-1)^{0} \binom{3}{0} + (-1)^{1} \binom{3}{0}\right] + \left[(-1)^{1} \binom{3}{1} + (-1)^{2}\binom{3}{1}\right]\\
+ &\left[(-1)^{2} \binom{3}{2} + (-1)^{3}\binom{3}{2}\right] + \left[(-1)^{3} \binom{3}{3} + (-1)^{4}\binom{3}{3}\right]
\end{aligned} \\
&= 0. \quad\blacksquare
\end{align*}

\end{document}

Addendum to address the OP's follow-up request. The revised solution gets by with a single align* environment plus a couple of different \phantom directives; one gets used four times, the other twice.

enter image description here

\documentclass{amsart}
\usepackage{amssymb} % for "\blacksquare" macro

\setlength{\oddsidemargin}{0.0in}
\setlength{\evensidemargin}{0.0in} 
\setlength{\textwidth}{6.1in}
\setlength{\topmargin}{0.0in} 
\setlength{\textheight}{9in}

%% Two "phantoms"
\newcommand\phA{\phantom{\bigg[}}
\newcommand\phB{\phantom{{}=(-1)^{0} \binom{3}{0}}}
\begin{document}

\begin{align*}
\sum_{i=0}^{4} (-1)^{i} \binom{4}{i}
&=\phA 
   (-1)^{0} \binom{4}{0} + (-1)^{1} \binom{4}{1} 
 + (-1)^{2} \binom{4}{2} + (-1)^{3} \binom{4}{3} + (-1)^{4} \binom{4}{4}\\
&=\phA 
   (-1)^{0} \binom{3}{0} 
 + \biggl[(-1)^{1} \binom{3}{0} + (-1)^{1}\binom{3}{1}\biggr]
 + \biggl[(-1)^{2} \binom{3}{1} + (-1)^{2}\binom{3}{2}\biggr]\\
&\phA\phB
 + \biggl[(-1)^{3} \binom{3}{2} + (-1)^{3}\binom{3}{3}\biggr] 
 + (-1)^{4}\binom{3}{3} \\
&= \biggl[(-1)^{0} \binom{3}{0} + (-1)^{1}\binom{3}{0}\biggr] 
 + \biggl[(-1)^{1} \binom{3}{1} + (-1)^{2}\binom{3}{1}\biggr]\\
&\phA\phB
 + \biggl[(-1)^{2} \binom{3}{2} + (-1)^{3}\binom{3}{2}\biggr] 
 + \biggl[(-1)^{3} \binom{3}{3} + (-1)^{4}\binom{3}{3}\biggr]\\
&= 0. \quad\blacksquare
\end{align*}

\end{document}
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