Editing Combination (section)

=== Example of counting combinations ===
As a specific example, one can compute the number of five-card hands possible from a standard fifty-two card deck as:<ref>{{harvnb|Mazur|2010|loc=p. 21}}</ref>

<math display="block"> \binom{52}{5} = \frac{52\times51\times50\times49\times48}{5\times4\times3\times2\times1} = \frac{311{,}875{,}200}{120} = 
2{,}598{,}960.</math>

Alternatively one may use the formula in terms of factorials and cancel the factors in the numerator against parts of the factors in the denominator, after which only multiplication of the remaining factors is required:
<math display="block">\begin{alignat}{2}
  \binom{52}{5}
    &= \frac{52!}{5!47!} \\[5pt]
    &= \frac{52\times51\times50\times49\times48\times\cancel{47!}}{5\times4\times3\times2\times\cancel{1}\times\cancel{47!}} \\[5pt]
    &= \frac{52\times51\times50\times49\times48}{5\times4\times3\times2} \\[5pt]
    &= \frac{(26\times\cancel{2})\times(17\times\cancel{3})\times(10\times\cancel{5})\times49\times(12\times\cancel{4})}{\cancel{5}\times\cancel{4}\times\cancel{3}\times\cancel{2}} \\[5pt]
    &= {26\times17\times10\times49\times12} \\[5pt]
    &= 2{,}598{,}960.
\end{alignat}</math>

Another alternative computation, equivalent to the first, is based on writing

<math display="block"> \binom{n}{k} = \frac { ( n - 0 ) }1 \times \frac { ( n - 1 ) }2 \times \frac { ( n - 2 ) }3 \times \cdots \times \frac { ( n - (k - 1) ) }k,</math>

which gives

<math display="block"> \binom{52}{5} = \frac{52}1 \times \frac{51}2 \times \frac{50}3 \times \frac{49}4 \times \frac{48}5 = 2{,}598{,}960.</math>

When evaluated in the following order, {{math|52 ÷ 1 × 51 ÷ 2 × 50 ÷ 3 × 49 ÷ 4 × 48 ÷ 5}}, this can be computed using only integer arithmetic. The reason is that when each division occurs, the intermediate result that is produced is itself a binomial coefficient, so no remainders ever occur.

Using the symmetric formula in terms of factorials without performing simplifications gives a rather extensive calculation:

<math display="block">
\begin{align}
 \binom{52}{5} &= \frac{n!}{k!(n-k)!} = \frac{52!}{5!(52-5)!} = \frac{52!}{5!47!} \\[6pt]
&= \tfrac{80,658,175,170,943,878,571,660,636,856,403,766,975,289,505,440,883,277,824,000,000,000,000}{120\times258,623,241,511,168,180,642,964,355,153,611,979,969,197,632,389,120,000,000,000} \\[6pt]
&= 2{,}598{,}960.
\end{align}</math>