From e3b60cbc093aaf1ce8e519c46872144abc2f77e3 Mon Sep 17 00:00:00 2001
From: newt <hi@newty.dev>
Date: Wed, 9 Oct 2024 18:02:38 +0100
Subject: [PATCH] feat(euler): 15 - Lattice paths

-
---
 euler/src/15 - Lattice paths.ts      | 14 ++++++++++++++
 euler/thoughts/15 - Lattice paths.md |  9 +++++++++
 2 files changed, 23 insertions(+)
 create mode 100644 euler/src/15 - Lattice paths.ts
 create mode 100644 euler/thoughts/15 - Lattice paths.md

diff --git a/euler/src/15 - Lattice paths.ts b/euler/src/15 - Lattice paths.ts
new file mode 100644
index 0000000..9ff61e2
--- /dev/null
+++ b/euler/src/15 - Lattice paths.ts	
@@ -0,0 +1,14 @@
+// See https://github.com/newtykins/the-honk/tree/main/euler/thoughts/15%20-Lattice%20Paths.md
+export {};
+
+const factorial = (n: number) => {
+    if (n < 0) return -1;
+    else if (n === 0) return 1;
+    else return n * factorial(n - 1);
+};
+
+const countLatticePaths = (width: number, height: number) => {
+    return factorial(width + height) / (factorial(width) * factorial(height));
+};
+
+console.log(countLatticePaths(20, 20));
diff --git a/euler/thoughts/15 - Lattice paths.md b/euler/thoughts/15 - Lattice paths.md
new file mode 100644
index 0000000..98fe247
--- /dev/null
+++ b/euler/thoughts/15 - Lattice paths.md	
@@ -0,0 +1,9 @@
+For a lattice with dimensions m x n, all paths will have m + n segments. m segments go down, n segments go right. To count m + n slots, count the ways to pick m downward moves - the rest will be rightward moves.
+
+![](assets/15.svg)
+
+### LaTeX
+
+```
+{m + n \choose m} = {m + n \choose n} = \frac{(m+n)!}{m!n!}
+```