How many set partitions are there of a 100-element set such that
(i) There are exactly 7 components, and each component has odd size

(ii) There are exactly an odd number of components, but the sizes of the components can be any strictly positive integer

(iii) There are exactly an odd number of components, but the sizes of the components can be any strictly positive integer except that we do not allow componets of size 2 and 5

How many permutations are there of {1,2,..., 100} such that
(i) There are exactly 7 cycles, and each cycle has odd size

(ii) There are exactly an odd number of cycles, but the length of the cycles can be any strictly positive integer

(iii) There are exactly an odd number of cycles, but the lengths of the cycles can be any strictly positive integer except that we do not allow cycles of length 2 and 5

Question 3
How many labeled connected graphs are there with 100 vertices? How many are there with exactly 2 components? With exactly 100 components?

Answer 3
Connected graphs
125452273648412141401139200773089339811437122262323352216462286700734387627331522885709312904647210371328794107858418293356140610389233010890575674923332327423429917033308833625087865757067009121643735335562957720919464360279849562024268481514680519466433342816295305461792608710076067024418040646132922593872156786882042514851134725907407734998659047116141535192232282791034794601497425933135733739224548153792567839779141913680549722028773475603674523853169051845195421079685934332310882053907313995168923535704556709136009686053259752740308577712484995569783650846269720521608249582420017860918845622644663874987575160515268766971368721963159046529551370104575096202684311358624098672672476994761800126366704378281769378475834662925454186722285874859694872242026068795039642380038691501013387521486053366727435218916045301026547623117868196574350026663672938544838421549439678291079144887169351257353939537143880911795509769485297854345823362270660712007071953958048732807833151699233932553517365602355625760928908188344900652899469143373311896574182769657272933821876553379518700197938172155010185719146936187180903250543579433005278984692348282568379624191843385007057047580846112172322370992146748764340302424100769640802926078370838482052406435490036434092467614465282934547622486501083341846045800849550829377015688082924233744178977266360888077466658752803757886543953371541948150838050857811138739570190403175194723845806878286855221273591078542964296886759202533689916185534005248

2 components
19792878830464097434146485069739463865673187370672887935834214122103592997909477467639145015413302778571558698199938989875722363632867076525500775193313203219625694859620492756307560817786124480904571667305985231399040766859432538780170724152051889531424530851027515639435541861971440089962657458762079240342770560135993633520904674629099681471079341302098763093130534058092714306341148454323169655736417188716167485638876856164406937663195315675056473320830072088996900023615988558834190601024551606490996047256677952803727152702099973371018028710701366236803289614033811563475696670364580819031243080945830466884602280114938472440443402744527681562033913068944126647000531086090506590597863434313244427618995892382119045374120085198882365703112462192132262796290715887651629501613500765319382697459714625232601391113377456170319146465206446792331217136689657731214890373782128820154293953635221308380687242012269185806693761170244502543846560216890121849276503785641853915478872687428718708183133628359872764435735322526121955322100244200579269171473679314961687680638065276652910066401714258899512328773662068014373263170965092624777284239415572512416129024249860116530210873861235283964287841875225918264476127609182676153172047392096718128978027994944226801334642136592881198292406724597930004937454732160477796576767954616879358825917759244042876084345101045377871469937328841023216205534951719728974096997883111102605252071058831339598818474925540591534080

100 components
only 1

I have trouble understanding exclusion in EGF's I will be coming to office hours on monday.