I am now going to do some of the same alignments in  \AmSTeX. Note
the subtle differences in equation labeling.

Here is an example of the align environment.
\TagsOnLeft

$$
\align
(a+b)^{n+1}
 &=(a+b)(a+b)^n=(a+b)\sum_{j=0}^n\binom nja^{n-1}b^j\tag1\\
 &=\sum_{j=0}^n\binom nj a^{n+1-j}b^j+\sum_{j=1}^n\binom
  n{j-1}a^{n-j}b^j\tag2\\
 &=\sum_{j=0}^n\binom{n+1}ja^{n+1-j}b^j\tag3
\endalign
$$

\TagsOnRight

$$
\align
(a+b)^{n+1}
 &=(a+b)(a+b)^n=(a+b)\sum_{j=0}^n\binom nja^{n-1}b^j\tag1\\
 &=\sum_{j=0}^n\binom nj a^{n+1-j}b^j+\sum_{j=1}^n\binom
  n{j-1}a^{n-j}b^j\tag2\\
 &=\sum_{j=0}^n\binom{n+1}ja^{n+1-j}b^j\tag3
\endalign
$$

Here is an example of the aligned environment.
\TagsOnLeft

$$
\aligned
(a+b)^{n+1}
 &=(a+b)(a+b)^n=(a+b)\sum_{j=0}^n\binom nja^{n-1}b^j\\
 &=\sum_{j=0}^n\binom nj a^{n+1-j}b^j+\sum_{j=1}^n\binom
  n{j-1}a^{n-j}b^j\\
 &=\sum_{j=0}^n\binom{n+1}ja^{n+1-j}b^j
\endaligned\tag5--4
$$

\TagsOnRight

$$
\aligned
(a+b)^{n+1}
 &=(a+b)(a+b)^n=(a+b)\sum_{j=0}^n\binom nja^{n-1}b^j\\
 &=\sum_{j=0}^n\binom nj a^{n+1-j}b^j+\sum_{j=1}^n\binom
  n{j-1}a^{n-j}b^j\\
 &=\sum_{j=0}^n\binom{n+1}ja^{n+1-j}b^j
\endaligned\tag5--4
$$

Here is an example of the split environment.
\TagsOnLeft

$$
\split
(a+b)^{n+1}
 &=(a+b)(a+b)^n=(a+b)\sum_{j=0}^n\binom nja^{n-1}b^j\\
 &=\sum_{j=0}^n\binom nj a^{n+1-j}b^j+\sum_{j=1}^n\binom
  n{j-1}a^{n-j}b^j\\
 &=\sum_{j=0}^n\binom{n+1}ja^{n+1-j}b^j
\endsplit\tag1--2
$$

\TagsOnRight

$$
\split
(a+b)^{n+1}
 &=(a+b)(a+b)^n=(a+b)\sum_{j=0}^n\binom nja^{n-1}b^j\\
 &=\sum_{j=0}^n\binom nj a^{n+1-j}b^j+\sum_{j=1}^n\binom
  n{j-1}a^{n-j}b^j\\
 &=\sum_{j=0}^n\binom{n+1}ja^{n+1-j}b^j
\endsplit\tag1--2
$$

Here is an example of the gather environment.
\TagsOnLeft

$$
\gather
(a+b)^{n+1}
 =(a+b)(a+b)^n=(a+b)\sum_{j=0}^n\binom nja^{n-1}b^j\tag1\\
 =\sum_{j=0}^n\binom nj a^{n+1-j}b^j+\sum_{j=1}^n\binom
  n{j-1}a^{n-j}b^j\tag2\\
 =\sum_{j=0}^n\binom{n+1}ja^{n+1-j}b^j\tag3
\endgather
$$

\TagsOnRight

$$
\gather
(a+b)^{n+1}
 =(a+b)(a+b)^n=(a+b)\sum_{j=0}^n\binom nja^{n-1}b^j\tag1\\
 =\sum_{j=0}^n\binom nj a^{n+1-j}b^j+\sum_{j=1}^n\binom
  n{j-1}a^{n-j}b^j\tag2\\
 =\sum_{j=0}^n\binom{n+1}ja^{n+1-j}b^j\tag3
\endgather
$$

Here is an example of the multline environment.
\TagsOnLeft

$$
\multline
(a+b)^{n+1}
 =(a+b)(a+b)^n\\
 =(a+b)\sum_{j=0}^n\binom nja^{n-1}b^j
 =\sum_{j=0}^n\binom nj a^{n+1-j}b^j+\sum_{j=1}^n\binom
  n{j-1}a^{n-j}b^j
 =\sum_{j=0}^n\binom{n+1}ja^{n+1-j}b^j
\endmultline\tag1--2
$$

\TagsOnRight

$$
\multline
(a+b)^{n+1}
 =(a+b)(a+b)^n\\
 =(a+b)\sum_{j=0}^n\binom nja^{n-1}b^j
 =\sum_{j=0}^n\binom nj a^{n+1-j}b^j+\sum_{j=1}^n\binom
  n{j-1}a^{n-j}b^j
 =\sum_{j=0}^n\binom{n+1}ja^{n+1-j}b^j
\endmultline\tag1--2
$$

\bye