l13_counting1

l13_counting1 - 6.042/18.062J Mathematics for Computer...

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Unformatted text preview: 6.042/18.062J Mathematics for Computer Science March 25, 2005 Srini Devadas and Eric Lehman Lecture Notes Counting I 20480135385502964448038 3171004832173501394113017 5763257331083479647409398 8247331000042995311646021 489445991866915676240992 3208234421597368647019265 5800949123548989122628663 8496243997123475922766310 1082662032430379651370981 3437254656355157864869113 6042900801199280218026001 8518399140676002660747477 1178480894769706178994993 3574883393058653923711365 6116171789137737896701405 8543691283470191452333763 1253127351683239693851327 3644909946040480189969149 6144868973001582369723512 8675309258374137092461352 1301505129234077811069011 3790044132737084094417246 6247314593851169234746152 8694321112363996867296665 1311567111143866433882194 3870332127437971355322815 6814428944266874963488274 8772321203608477245851154 1470029452721203587686214 4080505804577801451363100 6870852945543886849147881 8791422161722582546341091 1578271047286257499433886 4167283461025702348124920 6914955508120950093732397 9062628024592126283973285 1638243921852176243192354 4235996831123777788211249 6949632451365987152423541 9137845566925526349897794 1763580219131985963102365 4670939445749439042111220 7128211143613619828415650 9153762966803189291934419 1826227795601842231029694 4815379351865384279613427 7173920083651862307925394 9270880194077636406984249 1843971862675102037201420 4837052948212922604442190 7215654874211755676220587 9324301480722103490379204 2396951193722134526177237 5106389423855018550671530 7256932847164391040233050 9436090832146695147140581 2781394568268599801096354 5142368192004769218069910 7332822657075235431620317 9475308159734538249013238 2796605196713610405408019 5181234096130144084041856 7426441829541573444964139 9492376623917486974923202 2931016394761975263190347 5198267398125617994391348 7632198126531809327186321 9511972558779880288252979 2933458058294405155197296 5317592940316231219758372 7712154432211912882310511 9602413424619187112552264 3075514410490975920315348 5384358126771794128356947 7858918664240262356610010 9631217114906129219461111 3111474985252793452860017 5439211712248901995423441 7898156786763212963178679 9908189853102753335981319 3145621587936120118438701 5610379826092838192760458 8147591017037573337848616 9913237476341764299813987 3148901255628881103198549 5632317555465228677676044 8149436716871371161932035 3157693105325111284321993 5692168374637019617423712 8176063831682536571306791 Two different subsets of the ninety 25-digit numbers shown above have the same sum. For example, maybe the sum of the numbers in the first column is equal to the sum of the numbers in the second column. Can you find two such subsets? This is a very difficult computational problem. But well prove that such subsets must exist! This is the sort of weird conclusion one can reach by tricky use of counting, the topic of this chapter....
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l13_counting1 - 6.042/18.062J Mathematics for Computer...

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