Project Euler(53)

Posted on Saturday, June 27th, 2009 at 7:31 pm by Universe Queen

Problem 53:

There are exactly ten ways of selecting three from five, 12345:

123, 124, 125, 134, 135, 145, 234, 235, 245, and 345

In combinatorics, we use the notation, ^(5)C_(3) = 10.

In general,
^(n)C_(r) =
n!

r!(n−r)!
,where r ≤ n, n! = n×(n−1)×…×3×2×1, and 0! = 1.

It is not until n = 23, that a value exceeds one-million: ^(23)C_(10) = 1144066.

How many, not necessarily distinct, values of ^(n)C_(r), for 1 ≤ n ≤ 100, are greater than one-million?

Answer: 4075

Implementation in Java:

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import java.math.*;
 
public class Euler53 
{
	public static BigInteger multiply( int from, int to )
	{
		BigInteger result = BigInteger.valueOf(from);
 
		if( from == to )
		{
			return BigInteger.valueOf(1);
		}
 
		for( int i=from+1; i<=to; i++ )
		{
			result = result.multiply( BigInteger.valueOf(i) );
		}
 
		return result;
	}
 
	public static BigInteger c( int n, int r )
	{
		if( n == r )
		{
			return BigInteger.valueOf(1);
		}
 
		BigInteger l1 = multiply( r+1, n );
		BigInteger l2 = multiply( 1, n-r );
 
		return l1.divide( l2 );
	}
 
	public static void main(String[] args)
	{
		long result = 0;
 
		for( int n=1; n<=100; n++ )
		{
			for( int r=1; r<=n; r++ )
			{
				if( c( n, r ).compareTo( BigInteger.valueOf(1000000) ) > 0 )
				{ 
					result++;
				}
			}
		}
 
		System.out.println( result );
	}
}

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