Well this gives me the opportunity to use the inclusion/exclusion principle, used in my Dice Proposition puzzle.

It's easier to explain using ternary numbers (fewer digits to worry about).

There, p(5), say, represents the probability of getting all three possible digits in what amounts to five tries. I'll use capital P to designate the probability of a given set, of 1, 2 or 3 of the digits, with the digits specified: for example P(0 and 2) would be the probability of getting both at least one zero and at least one 2. P(0 or 1) would be the probability that at least one of those two digits would appear among the five specified for p(5).

In that terminology p(5) is the equivalent of P(0 and 1 and 2) in this ternary problem.

Using the inclusion/exclusion method:

P(0 and 1 and 2) = P(0) + P(1) + P(2) - P(0 or 1) - P(0 or 2) - P(1 or 2) + P(0 or 1 or 2)

Since the probabilities of each possible digit are equal, the terms can be combined:

3*P(0) - 3*P(0 or 1) + P(0 or 1 or 2)

or more generally

C(3,1)*P(0) - C(3,2)*P(0 or 1) + C(3,3)*P(0 or 1 or 2)

Here, remember that P(0 or 1), for example is equivalent to the probability that not all the digits are 2, and P(0) is the probabilty that not all the digits come from the set {1,2}.

Numerically that's 3*(1-(2/3)^5) - 3*(1-(1/3)^5) + 1, which comes out to about .61728395061729.

Back to Decimal:

p(n) =

10*P(0) - C(10,2)*P(0 or 1) + C(10,3)*P(0 or 1 or 2) - ... + C(10,9)*P(0 or 1 or 2 or 3 or 4 or 5 or 6 or 7 or 8) - P(0 or 1 or 2 or 3 or 4 or 5 or 6 or 7 or 8 or 9) 

where capital P(i) is defined contingent on n so that the following substitution represents the case for n:

10*(1-(9/10)^n) - C(10,2)*(1-(8/10)^n) + C(10,3)*(1-(7/10)^n) - ... + C(10,9)*(1-(1/10)^n) - 1

    4   kill "pandig+.txt"

    5   open "pandig+.txt" for output as #2

   10   for N=1 to 100

   20   Tot=0

   30   for I=1 to 10

   40      Term=combi(10,I)*(1-((10-I)//10)^N)*(-1)^(I-1)

   50      Tot=Tot+Term

   60   next

   70   print N,Tot,Tot/1

   75   print #2,N,Tot,Tot/1

   80   next

finds

               probability

  n   rational            decimal approximation

   1  0   0 

   2  0   0 

   3  0   0 

   4  0   0 

   5  0   0 

   6  0   0 

   7  0   0 

   8  0   0 

   9  0   0 

  10  567/1562500   0.00036288 

  11  6237/3125000   0.00199584 

  12  193347/31250000   0.006187104 

  13  891891/62500000   0.014270256 

  14  26675649/976562500   0.027315864576 

  15  143594451/(3125x10^6)   0.04595022432 

  16  10985907933/(15625x10^7)   0.0703098107712 

  17  31279509261/(3125x10^8)   0.1000944296352 

  18  21042596876301/(15625x10^10)   0.1346726200083263999 

  19  54125483631219/(3125x10^11)   0.1732015476199007999 

  20  671054134991877/(3125x10^12)   0.2147373231974006399 

  21  1614524160366117/(625x10^13)   0.2583238656585787199 

  22  236763267123649959/(78125x10^13)   0.3030569819182719474 

  23  271972926778836591/(78125x10^13)   0.3481253462769108364 

  24  122760128744112321/(3125x10^14)   0.3928324119811594271 

  25  54575491469075577/(125x10^15)   0.4366039317526046159 

  26  1496829579280832563767/(3125x10^18)   0.4789854653698664203 

  27  3247709559153727376709/(625x10^19)   0.5196335294645963802 

  28  34893950629539961690731/(625x10^20)   0.558303210072639387 

  29  74354284708591053185019/(125x10^21)   0.5948342776687284254 

  30  24575671455185092431306663/(390625x10^20)   0.6291371892527383661 

  31  103309351627654392774534981/(15625x10^22)   0.6611798504169881137 

  32  1079649425698155662929610031/(15625x10^23)   0.6909756324468196242 

  33  2245540231325863045575464463/(3125x10^24)   0.7185728740242761745 

  34  1162571749958790105933707292969/(15625x10^26)   0.7440459199736256677 

  35  479679774018898077792619889283/(625x10^27)   0.7674876384302369244 

  36  24656352941464094518259205701241/(3125x10^28)   0.7890032941268510245 

  37  50544101526286159123775820994617/(625x10^29)   0.8087056244205785459 

  38  6458679336808762545011618737341693/(78125x10^29)   0.8267109551115216057 

  39  3293500769359589784930588003763413/(390625x10^28)   0.8431361969560549849 

  40  33519397476006770622816892955319189/(390625x10^29)   0.8580965753857733279 

  41  68101871989013104864349691655490799/(78125x10^30)   0.8717039614593677422 

  42  138135263903957941276240328899531879503/(15625x10^34)   0.8840656889853308241 

  43  279776174841570667623827928185793700629/(3125x10^35)   0.8952837594930261363 

  44  565908970243105819274589689854474583919/(625x10^36)   0.9054543523889693108 

  45  228666892936369630216872051534797359443/(25x10^37)   0.9146675717454785208 

  46  36054975529202343291429654249040207702473/(390625x10^35)   0.9230073735475799882 

  47  290797383708258262753863345560109632558799/(3125x10^38)   0.9305516278664264408 

  48  2929288373226460095302724132118847017674861/(3125x10^39)   0.9373722794324672304 

  49  5897097360015813777241205577483673494626109/(625x10^40)   0.9435355776025302043 

  50  14829724264494253692571244783859722474239215237/(15625x10^42)   0.9491023529276322362 

  51  29816510081599368816476345831691507788276164587/(3125x10^43)   0.954128322611179802 

  52  299582628521801879031968833819882653723334636077/(3125x10^44)   0.9586644112697660128 

  53  601723172953395427160854315746092422549492792461/(625x10^45)   0.9627570767254326834 

  54  75503799469196254316909028388540898694196780839027/(78125x10^45)   0.9664486332057120552 

  55  15152774475381711824066695130847742737390601166961/(15625x10^45)   0.9697775664244295567 

  56  759983466143077997052562676602336355943974923595979/(78125x10^46)   0.9727788366631398362 

  57  1524194011336303865864543916024741090121240787675703/(15625x10^47)   0.9754841672552344741 

  58  1528003620098572469747040585608424677462026934794432971/(15625x10^50)   0.9779223168630863805 

  59  3062872920911427714160143540016402303785716667401174289/(3125x10^51)   0.9801193346916568684 

  60  30690587448369831810648637422738555333290988195411480447/(3125x10^52)   0.9820987983478346179 

  61  61492627154535185194682444376156870816095039389568564367/(625x10^53)   0.983882034472562963 

  62  3849563760055779018423888122998663987861595575272161210397/(390625x10^52)   0.9854883225742794286 

  63  15420860667233163144122093150097566437352582520301248342897/(15625x10^54)   0.9869350827029224412 

  64  30882438992008591800652193686632601253481233197645983870039/(3125x10^55)   0.9882380477442749376 

  65  12367642765013658147960547213936516505761769048673282264939/(125x10^56)   0.9894114212010926518 

  66  30952125667916151347652825038745285994698675525593268606920301/(3125x10^58)   0.9904680213733168431 

  67  61963713304061252906501668877187313990408810459942302936560667/(625x10^59)   0.9914194128649800464 

  68  620172516461414280878220098987189726562180164644434169796804973/(625x10^60)   0.9922760263382628493 

  69  1241309084264834707063976905774938580510294028512699832847097197/(125x10^61)   0.9930472674118677656 

  70  776360637161210011569348038755878790557990632599566471718550833161/(78125x10^61)   0.9937416155663488148 

  71  48553062201027986569649583459600376219624851865925570855685492033/(48828125x10^57)   0.9943667138770531648 

  72  1943221582712465025180055430841460742343296169092972217613506255087/(1953125x10^60)   0.9949294503487820928 

  73  486052749793625443460238141801102045762777968601067397476391159457/(48828125x10^58)   0.9954360315773449081 

  74  15560813272085758656152934738764613075693942325665813229037651394165639/(15625x10^66)   0.9958920494134885539 

  75  1245378176569287244601772318529906043642580080295812287996816015480917/(125x10^67)   0.9963025412554297956 

  76  311460013921328164004742264887135051550804965499722491506573892512303891/(3125x10^68)   0.9966720445482501247 

  77  623127903763954732975497328227737681907623668844052569558914017909271107/(625x10^69)   0.9970046460223275727 

  78  2434824282615148985102871339760643873097879704609285733228672475094255079/(244140625x10^64)   0.9973040261591650242 

  79  155870859270573193730735158341158902539786230611753571173285841714737716087/(15625x10^70)   0.9975734993316684398 

  80  1559087578167581477410242265249402465532199448684558163257815006030968506341/(15625x10^71)   0.9978160500272521455 

  81  3118857392269179984780497784963845338279540599566101325059742251607585304021/(3125x10^72)   0.9980343655261375951 

  82  1559735727146624250422692767135119883142884336181663247316287401482624374386173/(15625x10^74)   0.9982308653738395202 

  83  3120024149862584753291799134115108792006648060076625256282728729436810882101859/(3125x10^75)   0.998407727956027121 

  84  6241043215347530958786922244672573448003202475965576024866497313986687696922897/(625x10^76)   0.9985669144556049533 

  85  499355095222927602184019053985670668407797620557677957307886017554287735384793/(5x10^77)   0.9987101904458552043 

  86  1560686164608882340629043048644276246171023143788384288917639705387870129668252659/(15625x10^77)   0.998839145349684698 

  87  1560867515583257392325064973658864851726585636112786637570011269480594772737276479/(15625x10^77)   0.998955209973284731 

  88  15610307379729634441449575845792835060959470521078901563634318861732863447274356121/(15625x10^78)   0.9990596723026966042 

  89  31223552866679346922398561835991468287229206703981781552709558833568342742924416389/(3125x10^79)   0.9991536917337391014 

  90  156130986232712317800431094347137986965239208162190145774908346305129614156226444779907/(15625x10^82)   0.9992383118893588339 

  91  312285772551118314382250816660005290582544251197782844935661318930550964519013871246537/(3125x10^83)   0.9993144721635786059 

  92  3123071931617868090015533528220035903024656187906088298468417582662408894161394433196007/(3125x10^84)   0.9993830181177177887 

  93  6246529442764527549444183654063284781039499904355095745266521457615398154823411275711031/(625x10^85)   0.9994447108423244078 

  94  390429779448117663983046172038709227051591252165642876639931868328334413471488425342238931/(390625x10^84)   0.9995002353871812197 

  95  312359440109914085610780435959285776632918885643497418874958190571430128093987019686589753/(3125x10^86)   0.9995502083517250739 

  96  15618674761236805457242649217139773214183312153429255487282348831554532901699344883918950583/(15625x10^87)   0.9995951847191555492 

  97  31238614500291180054846938278735423200763473426764065223999398182713381537788580416096913751/(3125x10^88)   0.9996356640093177617 

  98  15619876497156952599857992009868641221497962270826043293811996266215125205667186851401624559641/(15625x10^90)   0.9996720958180449663 

  99  31240777650123918475441450858123526253613737113992198554375639903423969118715106504926227480959/(3125x10^91)   0.9997048848039653911 

 100   312416998493046001572329000213994444808820158829131349249531876637988573456837082185196018390217/(3125x10^92)   0.999734395177747205 

(Strings of zeros converted to powers of 10 by a separate QB program)

 

So the answers:

 

a: 27   3247709559153727376709/6250000000000000000000   0.5196335294645963802 

b: 35   479679774018898077792619889283/625000000000000000000000000000   0.7674876384302369244 

c: 51     0.954128322611179802

for those curious, the numerator and denominator, reduced to lowest terms, for p(51) are:

29816510081599368816476345831691507788276164587

31250000000000000000000000000000000000000000000