How to handle long equation in LaTeX?

Question

I used the code below, but the equation does not fit the page as it is too long. I tried to reduce the font size but I could not. What could be done to fit the equation to the line?

I am also wondering why the equation does not start from the beginning of line. As you can see from the pic below, there is huge space in the left side of the equation

\documentclass{report}
\usepackage{amsmath}

\begin{document}

\begin{align}
 {{f}_{r:n}}(x)=\frac{n!}{(r-1)!(n-r+1)!}\,\left[ {{\left( \frac{x-1}{k}\right)}^{r-1}}\ {{\left( \frac{k-x+1}{k} \right)}^{n-r+1}}-{{\left( \frac{x}{k} \right)}^{r-1}}\ {{\left( \frac{k-x}{k} \right)}^{n-r+1}} \right]+{{f}_{r-1:n}}(x)                    
\end{align}

\end{document}

Here is the output

Answer 1

I suggest you embed a split environment (provided by the amsmath package) inside an equation environment.

Aside: Don't overuse { and } in math mode, and don't overuse auto-sizing of round parentheses and square brackets. Do take the time to look at the equation and fine-tune some of the horizontal spacing; in the code below, you'll find two instances of \, (positive thinspace) and four instances of \! (negative thinspace).

\documentclass{article}
\usepackage{amsmath} % for "split" environment
\begin{document}

\begin{equation}\begin{split}
f_{r:n}(x)&=\frac{n!}{(r-1)!\,(n-r+1)!}
   \Biggl[ \biggl( \frac{x-1}{k}  \biggr)^{\!r-1} 
           \biggl( \frac{k-x+1}{k}\biggr)^{\!n-r+1} \\
  &\qquad -\biggl( \frac{x}{k}    \biggr)^{\!r-1} 
           \biggl( \frac{k-x}{k}  \biggr)^{\!n-r+1} \,
   \Biggr]+ f_{r-1:n}(x)                    
\end{split}\end{equation}

\end{document} 

Answer 2

Or use multline:

\documentclass{article}
\usepackage{amsmath}
\begin{document}

\begin{multline}
  f_{r:n}(x)=\frac{n!}{(r-1)!\,(n-r+1)!}
    \Biggl[ \biggl( \frac{x-1}{k}   \biggr)^{\!r-1}
            \biggl( \frac{k-x+1}{k} \biggr)^{\!n-r+1} \\
          - \biggl( \frac{x}{k}     \biggr)^{\!r-1}
            \biggl( \frac{k-x}{k}   \biggr)^{\!n-r+1} \,
    \Biggr] + f_{r-1:n}(x)
\end{multline}

\end{document}

Answer 3

Answer reversed to show the 'logical' solution first, and the 'literal' answer second

If you have an equation which takes the length of a page, making it smaller / narrower / splitting onto several lines etc, is unlikely to aid the reader that much, as it is likely to be very dense and hard to read in the first place. If you have an equation like that, it probably means you're doing something wrong, and it would be worth considering if your equation can be restated in a more digestible way, as logical steps.

I think your equation could easily be restated, e.g. as:

      

f_{r:n}(x) = \binom {n} {r \! - \! 1} 
             \biggl [ \Phi(x \! - \! 1) - \Phi(x) \biggr ] 
             + f_{r-1:n}(x)

\Phi(x) = \biggl ( \frac {x}   {k} \biggr ) ^ {r-1} 
          \biggl ( \frac {k-x} {k} \biggr ) ^ {n - r + 1}

which both saves you space and makes it clearer what's going on.

However, to answer your question literally, if you really don't want to split in this way, and you're just asking if it is at all possible with some sort of 'hack' to fit it all into one page / line, then you can save some space by making use of the 'choose r from n' notation, maybe substitute r-1 and x-1 with something else, and generally play with mkern to remove unnecessary extra space you don't want, particularly space resulting from the superscripts, e.g.:

f_{r:n}(x) = \binom{n}{r'} 
  \mkern -4mu  \left [
  \mkern -4mu    \left ( \frac {x'}     {k} \right ) ^ { \!\! r'   }
  \mkern -12mu   \left ( \frac {k - x'} {k} \right ) ^ { \!\! n-r' }
  \mkern -24mu - \left ( \frac {x}      {k} \right ) ^ { \!\! r'   }
  \mkern -12mu   \left ( \frac {k-x}    {k} \right ) ^ { \!\! n-r' }
  \right ] \! + \! f_{r':n}(x)

\text{where:}~~r' = r - 1, ~~~x' = x - 1

However, as you might gather from the controversy this latter approach has managed to gather in the comments below, this just packs everything on one line, is probably not the best idea anyway, and some people feel quite strongly about it.