From e5b47cd56b6ec2446f3ee98875fe1dc578fc1e72 Mon Sep 17 00:00:00 2001 From: mark Date: Sun, 5 Nov 2023 20:09:04 -0800 Subject: [PATCH] Added a solution --- Intermediate/Combinatorics/main.tex | 16 +++++++++++++++- 1 file changed, 15 insertions(+), 1 deletion(-) diff --git a/Intermediate/Combinatorics/main.tex b/Intermediate/Combinatorics/main.tex index 5c14473..8249dc5 100755 --- a/Intermediate/Combinatorics/main.tex +++ b/Intermediate/Combinatorics/main.tex @@ -246,7 +246,21 @@ \vfill \problem{} - How many ways can we split the number 2016 into a sum of positive integers? + How many ways can we split the number 2016 into a sum of positive integers? \par + \note{Consider $2016 + 1$ and $1 + 2016$ distinct sums. Order matters.} + + \begin{solution} + Split 2016 into ones, and put a \say{bit} between each pair. \par + This gives us $2^{2015}$ positions to place a bar, and thus $2^{2016}$ possible sums. + + \vspace{2mm} + + You could also sum over the usual stars-and-bars technique to get the same result. \par + Showing that they're equal could be a good bonus problem! + \end{solution} + + + \vfill