In JavaScript Numbers Can Bite, I talked about a problem that can crop up in JavaScript when you’re doing calculations with integers.
At the end of the post I hinted at a better way to write the combination function. Let’s go ahead and implement it. I’ll throw a permutation function in as well.
Let’s look at combinations. The formula is:
n!/(k!*(n-k)!)
That’s for n items, taken k at a time. In my mom’s family, there were seven children. How many combinations of them are there if you take 4 at a time?
7!/(4!*3!)
Now, if we write out the factorials, we immediately see a simplification (not in notation, but in floating point operations).
(7*6*5*4*3*2*1)/(4*3*2*1*3*2*1) = (7*6*5)/(3*2*1)
What we need here is a function that multiplies a range of numbers. Then we have:
productRange(5,7)/productRange(1,3)
or more generally,
productRange(k+1,n)/productRange(1,n-k)
function productRange(a,b) { var product=a,i=a; while (i++<b) { product*=i; } return product; } function combinations(n,k) { if (n==k) { return 1; } else { k=Math.max(k,n-k); return productRange(k+1,n)/productRange(1,n-k); } }
So that’s combinations, where the order of the items doesn’t matter. What about permutations, where the order does matter? The formula is:
n!/(n-k)!
That one is simple…
function permutations(n,k) { return productRange(k+1,n); }