mscroggs.co.uk
mscroggs.co.uk

subscribe

Comment

Comments

Comments in green were written by me. Comments in blue were not written by me.
You can solve the Dec 21 puzzle using the principle of inclusion/exclusion:

-There are 6! total ways of arranging 6 numbers.
-Now we have to exclude the ones that don't fit. How many ways have 2 following 1? You can think of 12 as a pair, so you're arranging 12/3/4/5/6 in any order, so there are 5! ways to do this. And there are (5 choose 1)=5 total pairs that might exist, so there are 5*5! ways that have either 12, 23, 34, 45, or 56.
-Of course, we've double counted some that have more than one pair. (This is where inclusion/exclusion comes in, we have to include them back in). So how many have, say, 12 and 45? Well now we're arranging 12/3/45/6, so there are 4! ways to do so. There are (5 choose 2)=10 different pairs, so the double counting was 10*4!.
-We continue this on, and inclusion/exclusion says we keep alternating adding and subtracting as we add more pairs, so the answer is:
6!
- (5 choose 1) * 5!
+ (5 choose 2) * 4!
- (5 choose 3) * 3!
+ (5 choose 4) * 2!
- (5 choose 5) * 1!
= 309
Todd
on /blog/88
×3   ×1            

Archive

Show me a random blog post
 2024 

Feb 2024

Zines, pt. 2

Jan 2024

Christmas (2023) is over
 2023 
▼ show ▼
 2022 
▼ show ▼
 2021 
▼ show ▼
 2020 
▼ show ▼
 2019 
▼ show ▼
 2018 
▼ show ▼
 2017 
▼ show ▼
 2016 
▼ show ▼
 2015 
▼ show ▼
 2014 
▼ show ▼
 2013 
▼ show ▼
 2012 
▼ show ▼

Tags

christmas game of life final fantasy latex coins recursion python golden spiral error bars talking maths in public bodmas draughts matrices data visualisation weak imposition chebyshev errors propositional calculus fractals anscombe's quartet gaussian elimination realhats news radio 4 edinburgh logs martin gardner pi numerical analysis probability tmip plastic ratio phd football big internet math-off javascript sound puzzles royal baby tennis the aperiodical flexagons platonic solids newcastle correlation national lottery crochet manchester science festival reuleaux polygons a gamut of games pascal's triangle inverse matrices polynomials standard deviation dates signorini conditions reddit advent calendar gather town rugby logic go christmas card curvature map projections video games fonts braiding approximation squares speed finite group world cup hyperbolic surfaces geogebra guest posts hats matrix multiplication hexapawn dataset chalkdust magazine oeis wool palindromes triangles rhombicuboctahedron graphs php matrix of cofactors preconditioning trigonometry noughts and crosses pizza cutting numbers statistics nine men's morris stirling numbers stickers inline code game show probability mathslogicbot mathsjam books exponential growth databet pythagoras frobel live stream youtube hannah fry royal institution determinants weather station craft folding tube maps cambridge dragon curves cross stitch arithmetic gerry anderson sorting people maths bubble bobble countdown crossnumber computational complexity quadrilaterals chess raspberry pi turtles zines london underground misleading statistics mean wave scattering machine learning pi approximation day interpolation captain scarlet graph theory pac-man mathsteroids simultaneous equations bempp matrix of minors datasaurus dozen asteroids dinosaurs logo menace binary 24 hour maths sport data ternary runge's phenomenon european cup fence posts electromagnetic field ucl matt parker boundary element methods estimation games folding paper sobolev spaces finite element method accuracy golden ratio harriss spiral geometry programming london light manchester convergence

Archive

Show me a random blog post
▼ show ▼
© Matthew Scroggs 2012–2024