Calculate and check the probability for at least one of m buckets to contain at least k of n uniformly randomly placed balls; see https://math.stackexchange.com/questions/2839811.

Question2839811.java

import java.math.BigInteger;

public class Question2839811 {
	final static int n = 180;
	final static int m = 10;
	final static int k = 24;

	// XOR shift pseudorandom number generator because Java's built-in generator has regularities that mess up the result
	static long x = 4;
	static long cache;
	static int ncached;
	
	static int nbits;
	static {
		for (int bits = m - 1;bits != 0;bits >>= 1)
			nbits++;
	}
	
	static long randomLong () {
		x ^= (x << 21);
		x ^= (x >>> 35);
		x ^= (x << 4);
		return x;
	}
	
	static int randomBucket () {
		for (;;) {
			if (ncached == 0) {
				cache = randomLong ();
				ncached = 64 / nbits;
			}
			long bucket = (cache & ((1L << nbits) - 1));
			cache >>= nbits;
			ncached--;
			if (bucket < m)
				return (int) bucket;
		}
	}

	final static int ntrials = 100000000;
	
	static BigInteger [] powers = new BigInteger [m + 1];

	public static void main (String [] args) {
		for (int i = 0;i <= m;i++)
			powers [i] = BigInteger.ONE;
		
		for (int j = 1;j <= n;j++)
			for (int i = 0;i <= m;i++)
				powers [i] = powers [i].multiply (BigInteger.valueOf (i));

		BigInteger factor = BigInteger.ONE;
		BigInteger sum = BigInteger.ZERO;
		
		for (int i = 0;i <= 3;i++) {
			factor = factor.negate ();
			if (i != 0) {
				BigInteger pi = recurse (i);
				sum = sum.add (pi.multiply (factor));
				System.out.println (i + " : " + pi.doubleValue () / powers [m].doubleValue ());
				System.out.println ("    " + pi);
				System.out.println ("    " + sum.doubleValue () / powers [m].doubleValue ());
				System.out.println ("    " + sum);
			}
			factor = factor.multiply (BigInteger.valueOf (m - i)).divide (BigInteger.valueOf (i + 1));
		}
		
		long count = 0;
		long [] counts = new long [4];
		outer:
		for (int j = 0;j < ntrials;j++) {
			int [] buckets = new int [m];
			for (int i = 0;i < n;i++)
				buckets [randomBucket ()]++;
			for (int b : buckets)
				if (b >= k) {
					count++;
					continue outer;
				}
//			for (int i = 0;i < 4;i++)
//				if (buckets [i] >= k)
//					counts [i]++;
//				else
//					continue outer;
		}
		
		for (int i = 0;i < 4;i++)
			System.out.println (i + " : " + counts [i] / (double) ntrials);

		System.out.println (count / (double) ntrials);
}
	
	static BigInteger recurse (int d) {
		return recurse (0,d,BigInteger.valueOf (m - d),powers [m - d]);
	}

	static BigInteger recurse (int indexSum,int d,BigInteger md,BigInteger product) {
		d--;
		
		BigInteger sum = BigInteger.ZERO;
		for (int i = 0;i <= n;i++,indexSum++) {
			if (i >= k)
				sum = sum.add (d == 0 ? product : recurse (indexSum,d,md,product));
			product = product.multiply (BigInteger.valueOf (n - indexSum));
			if (product.signum () == 0)
				break;
			product = product.divide (BigInteger.valueOf (i + 1));
			product = product.divide (md);
		}
		return sum;
	}
}