Editing Binomial coefficient (section)

=== Multiplicative formula ===
A more efficient method to compute individual binomial coefficients is given by the formula
<math display="block">\binom nk = \frac{n^{\underline{k}}}{k!} = \frac{n(n-1)(n-2)\cdots(n-(k-1))}{k(k-1)(k-2)\cdots 1} =  \prod_{i=1}^k\frac{ n+1-i}{ i},</math>
where the numerator of the first fraction, <math>n^{\underline{k}}</math>, is a [[falling factorial]].
This formula is easiest to understand for the combinatorial interpretation of binomial coefficients.
The numerator gives the number of ways to select a sequence of {{mvar|k}} distinct objects, retaining the order of selection, from a set of {{mvar|n}} objects. The denominator counts the number of distinct sequences that define the same {{mvar|k}}-combination when order is disregarded. This formula can also be stated in a recursive form. Using the "C" notation from above, <math>C_{n,k} = C_{n, k-1} \cdot (n-k+1) / k</math>, where <math>C_{n,0} = 1</math>. It is readily derived by evaluating <math>C_{n,k} / C_{n, k-1}</math> and can intuitively be understood as starting at the leftmost coefficient of the <math>n</math>-th row of [[Pascal's triangle]], whose value is always <math>1</math>, and recursively computing the next coefficient to its right until the <math>k</math>-th one is reached.

Due to the symmetry of the [[Pascal's triangle|binomial coefficients]] with regard to {{mvar|k}} and {{math|''n'' − ''k''}}, calculation of the above product, as well as the recursive relation, may be optimised by setting its upper limit to the smaller of {{mvar|k}} and {{math|''n'' − ''k''}}.