Project Euler（53）
Posted on Saturday, June 27th, 2009 at 7:31 pm by Universe QueenProblem 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:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 | 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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