Problem 15 - Starting in the top left corner in a 20 by 20 grid, how many routes are there to the bottom right corner?

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本文探讨了一个20x20网格中从左上角到右下角的不同路径数量问题，在不回头的情况下，通过组合数学原理计算可能的路径总数。文章提供了一种简单的方法来计算这一数值：利用排列组合公式 (m+n)!/(m!n!) 来求解。

题目:

Starting in the top left corner of a 22 grid, there are 6 routes (without backtracking) to the bottom right corner.

How many routes are there through a 2020 grid?

	/**
	 * 
	 * 1. from 0,0 to m,n, it needs m+n steps.
	 * 
	 * 2. u need m steps to forward right.
	 * 
	 * 3. so, it's a simple permutation (m+n)!/(m!n!)
	 * 
	 * 4. It just like that u have 20 same red balls and 20 same blue balls.
	 * how many permutation there are.
	 * 
	 * @param m
	 * @param n
	 * @return
	 */
	public static long findRoutes(int m, int n) {
		BigDecimal bd = BigDecimal.ONE;
		for (int i = n + 1; i <= m + n; i++) {
			bd = bd.multiply(new BigDecimal(i));
		}

		for (int i = 1; i <= m; i++) {
			bd = bd.divide(new BigDecimal(i));
		}
		System.out.println("----" + bd);
		return bd.longValue();
	}