When the number of outcomes is known, first decide whether outcomes are ordered. Translate to one permutation model or one combination model, then solve for the unknown. Cancel factorials before forming a polynomial. A number of available objects is an integer; for selecting or arranging r of n, require 0 <= r <= n. For a known pool, test the integer selection sizes systematically and use nr= nn-r. A valid smaller size may have a second, larger symmetric partner. Example: n2=15 gives n(n-1)=30, so (n-6)(n+5)=0. Only n=6 is a possible pool size. Also 64= 62=15. Check: do not discard a second positive selection-size solution merely because the first one works. Check each result in the original equation and its integer bounds.

When the number of outcomes is known, first decide whether outcomes are ordered. Translate to one permutation model or one combination model, then solve for the unknown.

Cancel factorials before forming a polynomial. A number of available objects is an integer; for selecting or arranging \(r\) of \(n\), require \(0\le r\le n\).

For a known pool, test the integer selection sizes systematically and use \(\binom nr=\binom n{n-r}\). A valid smaller size may have a second, larger symmetric partner.

Example: \(\binom n2=15\) gives \(n(n-1)=30\), so \((n-6)(n+5)=0\). Only \(n=6\) is a possible pool size. Also \(\binom64=\binom62=15\).

Check: do not discard a second positive selection-size solution merely because the first one works. Check each result in the original equation and its integer bounds.

Solving counting equations

Solving counting equations