Combinatoric 'N choose R' in java math?
Is there a built in method in a java library that can compute 'N choose R' for any N, R?
Solution 1:
The Formula
It's actually very easy to compute N choose K
without even computing factorials.
We know that the formula for (N choose K)
is:
N!
--------
(N-K)!K!
Therefore, the formula for (N choose K+1)
is:
N! N! N! N! (N-K)
---------------- = --------------- = -------------------- = -------- x -----
(N-(K+1))!(K+1)! (N-K-1)! (K+1)! (N-K)!/(N-K) K!(K+1) (N-K)!K! (K+1)
That is:
(N choose K+1) = (N choose K) * (N-K)/(K+1)
We also know that (N choose 0)
is:
N!
---- = 1
N!0!
So this gives us an easy starting point, and using the formula above, we can find (N choose K)
for any K > 0
with K
multiplications and K
divisions.
Easy Pascal's Triangle
Putting the above together, we can easily generate Pascal's triangle as follows:
for (int n = 0; n < 10; n++) {
int nCk = 1;
for (int k = 0; k <= n; k++) {
System.out.print(nCk + " ");
nCk = nCk * (n-k) / (k+1);
}
System.out.println();
}
This prints:
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
1 8 28 56 70 56 28 8 1
1 9 36 84 126 126 84 36 9 1
BigInteger
version
Applying the formula for BigInteger
is straightforward:
static BigInteger binomial(final int N, final int K) {
BigInteger ret = BigInteger.ONE;
for (int k = 0; k < K; k++) {
ret = ret.multiply(BigInteger.valueOf(N-k))
.divide(BigInteger.valueOf(k+1));
}
return ret;
}
//...
System.out.println(binomial(133, 71));
// prints "555687036928510235891585199545206017600"
According to Google, 133 choose 71 = 5.55687037 × 1038.
References
- Wikipedia/Binomial coefficient
- Wikipedia/Pascal's triangle
- Wikipedia/Combination
Solution 2:
The apache-commons "Math" supports this in org.apache.commons.math4.util.CombinatoricsUtils
Solution 3:
The recursive definition gives you a pretty simple choose function which will work fine for small values. If you're planning on running this method a lot, or on large values, it would pay to memoize it, but otherwise works just fine.
public static long choose(long total, long choose){
if(total < choose)
return 0;
if(choose == 0 || choose == total)
return 1;
return choose(total-1,choose-1)+choose(total-1,choose);
}
Improving the runtime of this function is left as an exercise for the reader :)
Solution 4:
I am just trying to calculate number of 2 card combinations with different deck sizes...
No need to import an external library - from the definition of combination, with n
cards that would be n*(n-1)/2
Bonus question: This same formula calculates the sum of the first n-1
integers - do you see why they're the same? :)