For standard labeled dice (examples: 2d6,5d6,6d6,3d4,4d4,5d8)
The number of ways and the probability to get V distinct die faces in K dice rolls
Max number of dice rolls: about 30. Your mileage will vary.
Xd6 data
\\\[ways, total ways, rational, decimal, percentage]
> distinctFaces(1,6,1)
[6 6 1 1.0000000 100.00000]
> distinctFaces(1,6,2)
[0 6 0 0 0]
> distinctFaces(1,6,3)
[0 6 0 0 0]
> distinctFaces(1,6,4)
[0 6 0 0 0]
> distinctFaces(1,6,5)
[0 6 0 0 0]
> distinctFaces(1,6,6)
[0 6 0 0 0]
> distinctFaces(2,6,1)
[6 36 1/6 0.16666667 16.666667]
> distinctFaces(2,6,2)
[30 36 5/6 0.83333333 83.333333]
> distinctFaces(2,6,3)
[0 36 0 0 0]
> distinctFaces(2,6,4)
[0 36 0 0 0]
> distinctFaces(2,6,5)
[0 36 0 0 0]
> distinctFaces(2,6,6)
[0 36 0 0 0]
> distinctFaces(3,6,1)
[6 216 1/36 0.027777778 2.7777778]
> distinctFaces(3,6,2)
[90 216 5/12 0.41666667 41.666667]
> distinctFaces(3,6,3)
[120 216 5/9 0.55555556 55.555556]
> distinctFaces(3,6,4)
[0 216 0 0 0]
> distinctFaces(3,6,5)
[0 216 0 0 0]
> distinctFaces(3,6,6)
[0 216 0 0 0]
> distinctFaces(4,6,1)
[6 1296 1/216 0.0046296296 0.46296296]
> distinctFaces(4,6,2)
[210 1296 35/216 0.16203704 16.203704]
> distinctFaces(4,6,3)
[720 1296 5/9 0.55555556 55.555556]
> distinctFaces(4,6,4)
[360 1296 5/18 0.27777778 27.777778]
> distinctFaces(4,6,5)
[0 1296 0 0 0]
> distinctFaces(4,6,6)
[0 1296 0 0 0]
> distinctFaces(5,6,1)
[6 7776 1/1296 0.00077160494 0.077160494]
> distinctFaces(5,6,2)
[450 7776 25/432 0.057870370 5.7870370]
> distinctFaces(5,6,3)
[3000 7776 125/324 0.38580247 38.580247]
> distinctFaces(5,6,4)
[3600 7776 25/54 0.46296296 46.296296]
> distinctFaces(5,6,5)
[720 7776 5/54 0.092592593 9.2592593]
> distinctFaces(5,6,6)
[0 7776 0 0 0]
> distinctFaces(6,6,1)
[6 46656 1/7776 0.00012860082 0.012860082]
> distinctFaces(6,6,2)
[930 46656 155/7776 0.019933128 1.9933128]
> distinctFaces(6,6,3)
[10800 46656 25/108 0.23148148 23.148148]
> distinctFaces(6,6,4)
[23400 46656 325/648 0.50154321 50.154321]
> distinctFaces(6,6,5)
[10800 46656 25/108 0.23148148 23.148148]
> distinctFaces(6,6,6)
[720 46656 5/324 0.015432099 1.5432099]
> distinctFaces(7,6,1)
[6 279936 1/46656 2.1433471 E-5 0.0021433471]
> distinctFaces(7,6,2)
[1890 279936 35/5184 0.0067515432 0.67515432]
> distinctFaces(7,6,3)
[36120 279936 1505/11664 0.12902949 12.902949]
> distinctFaces(7,6,4)
[126000 279936 875/1944 0.45010288 45.010288]
> distinctFaces(7,6,5)
[100800 279936 175/486 0.36008230 36.008230]
> distinctFaces(7,6,6)
[15120 279936 35/648 0.054012346 5.4012346]
> distinctFaces(8,6,1)
[6 1679616 1/279936 3.5722451 E-6 0.00035722451]
> distinctFaces(8,6,2)
[3810 1679616 635/279936 0.0022683756 0.22683756]
> distinctFaces(8,6,3)
[115920 1679616 805/11664 0.069015775 6.9015775]
> distinctFaces(8,6,4)
[612360 1679616 35/96 0.36458333 36.458333]
> distinctFaces(8,6,5)
[756000 1679616 875/1944 0.45010288 45.010288]
> distinctFaces(8,6,6)
[191520 1679616 665/5832 0.11402606 11.402606]
> distinctFaces(9,6,1)
[6 10077696 1/1679616 5.9537418 E-7 5.9537418 E-5]
> distinctFaces(9,6,2)
[7650 10077696 425/559872 0.00075910208 0.075910208]
> distinctFaces(9,6,3)
[363000 10077696 15125/419904 0.036020138 3.6020138]
> distinctFaces(9,6,4)
[2797200 10077696 6475/23328 0.27756344 27.756344]
> distinctFaces(9,6,5)
[5004720 10077696 11585/23328 0.49661351 49.661351]
> distinctFaces(9,6,6)
[1905120 10077696 245/1296 0.18904321 18.904321]
> distinctFaces(10,6,1)
[6 60466176 1/10077696 9.9229030 E-8 9.9229030 E-6]
> distinctFaces(10,6,2)
[15330 60466176 2555/10077696 0.00025353017 0.025353017]
> distinctFaces(10,6,3)
[1119600 60466176 7775/419904 0.018516137 1.8516137]
> distinctFaces(10,6,4)
[12277800 60466176 170525/839808 0.20305236 20.305236]
> distinctFaces(10,6,5)
[30618000 60466176 875/1728 0.50636574 50.636574]
> distinctFaces(10,6,6)
[16435440 60466176 38045/139968 0.27181213 27.181213]
> distinctFaces(11,6,1)
[6 362797056 1/60466176 1.6538172 E-8 1.6538172 E-6]
> distinctFaces(11,6,2)
[30690 362797056 1705/20155392 8.4592748 E-5 0.0084592748]
> distinctFaces(11,6,3)
[3420120 362797056 142505/15116544 0.0094270886 0.94270886]
> distinctFaces(11,6,4)
[52470000 362797056 364375/2519424 0.14462631 14.462631]
> distinctFaces(11,6,5)
[177645600 362797056 616825/1259712 0.48965557 48.965557]
> distinctFaces(11,6,6)
[129230640 362797056 99715/279936 0.35620642 35.620642]
> distinctFaces(12,6,1)
[6 2176782336 1/362797056 2.7563619 E-9 2.7563619 E-7]
> distinctFaces(12,6,2)
[61410 2176782336 10235/362797056 2.8211365 E-5 0.0028211365]
> distinctFaces(12,6,3)
[10383120 2176782336 24035/5038848 0.0047699395 0.47699395]
> distinctFaces(12,6,4)
[220140360 2176782336 3057505/30233088 0.10113109 10.113109]
> distinctFaces(12,6,5)
[993168000 2176782336 287375/629856 0.45625508 45.625508]
> distinctFaces(12,6,6)
[953029440 2176782336 1654565/3779136 0.43781568 43.781568]
> distinctFaces(13,6,1)
[6 13060694016 1/2176782336 4.5939366 E-10 4.5939366 E-8]
> distinctFaces(13,6,2)
[122850 13060694016 2275/241864704 9.4060851 E-6 0.00094060851]
> distinctFaces(13,6,3)
[31395000 13060694016 1308125/544195584 0.0024037773 0.24037773]
> distinctFaces(13,6,4)
[911710800 13060694016 6331325/90699264 0.069805693 6.9805693]
> distinctFaces(13,6,5)
[5406120720 13060694016 37542505/90699264 0.41392293 41.392293]
> distinctFaces(13,6,6)
[6711344640 13060694016 485485/944784 0.51385819 51.385819]
> distinctFaces(14,6,1)
[6 78364164096 1/13060694016 7.6565610 E-11 7.6565610 E-9]
> distinctFaces(14,6,2)
[245730 78364164096 40955/13060694016 3.1357445 E-6 0.00031357445]
> distinctFaces(14,6,3)
[94676400 78364164096 657475/544195584 0.0012081594 0.12081594]
> distinctFaces(14,6,4)
[3741028200 78364164096 17319575/362797056 0.047739017 4.7739017]
> distinctFaces(14,6,5)
[28854025200 78364164096 66791725/181398528 0.36820434 36.820434]
> distinctFaces(14,6,6)
[45674188560 78364164096 317181865/544195584 0.58284535 58.284535]
> distinctFaces(15,6,1)
[6 470184984576 1/78364164096 1.2760935 E-11 1.2760935 E-9]
> distinctFaces(15,6,2)
[491490 470184984576 27305/26121388032 1.0453120 E-6 0.00010453120]
> distinctFaces(15,6,3)
[285012120 470184984576 11875505/19591041024 0.00060617019 0.060617019]
> distinctFaces(15,6,4)
[15248142000 470184984576 35296625/1088391168 0.032430091 3.2430091]
> distinctFaces(15,6,5)
[151752182400 470184984576 43909775/136048896 0.32274995 32.274995]
> distinctFaces(15,6,6)
[302899156560 470184984576 233718485/362797056 0.64421274 64.421274]
> distinctFaces(16,6,1)
[6 2821109907456 1/470184984576 2.1268225 E-12 2.1268225 E-10]
> distinctFaces(16,6,2)
[983010 2821109907456 163835/470184984576 3.4844796 E-7 3.4844796 E-5]
> distinctFaces(16,6,3)
[857002320 2821109907456 5951405/19591041024 0.00030378197 0.030378197]
> distinctFaces(16,6,4)
[61847604360 2821109907456 858994505/39182082048 0.021923146 2.1923146]
> distinctFaces(16,6,5)
[789257196000 2821109907456 101499125/362797056 0.27976833 27.976833]
> distinctFaces(16,6,6)
[1969147121760 2821109907456 2279105465/3265173504 0.69800440 69.800440]
> distinctFaces(17,6,1)
[6 16926659444736 1/2821109907456 3.5447042 E-13 3.5447042 E-11]
> distinctFaces(17,6,2)
[1966050 16926659444736 109225/940369969152 1.1615109 E-7 1.1615109 E-5]
> distinctFaces(17,6,3)
[2574939000 16926659444736 107289125/705277476864 0.00015212328 0.015212328]
> distinctFaces(17,6,4)
[249961424400 16926659444736 1735843225/117546246144 0.014767322 1.4767322]
> distinctFaces(17,6,5)
[4069981188720 16926659444736 28263758255/117546246144 0.24044799 24.044799]
> distinctFaces(17,6,6)
[12604139926560 16926659444736 4862708305/6530347008 0.74463245 74.463245]
> distinctFaces(18,6,1)
[6 101559956668416 1/16926659444736 5.9078403 E-14 5.9078403 E-12]
> distinctFaces(18,6,2)
[3932130 101559956668416 655355/16926659444736 3.8717326 E-8 3.8717326 E-6]
> distinctFaces(18,6,3)
[7732681200 101559956668416 5966575/78364164096 7.6139075 E-5 0.0076139075]
> distinctFaces(18,6,4)
[1007570514600 101559956668416 13994034925/1410554953728 0.0099209427 0.99209427]
> distinctFaces(18,6,5)
[20849828792400 101559956668416 48263492575/235092492288 0.20529576 20.529576]
> distinctFaces(18,6,6)
[79694820748080 101559956668416 553436255195/705277476864 0.78470712 78.470712]
> distinctFaces(19,6,1)
[6 609359740010496 1/101559956668416 9.8464004 E-15 9.8464004 E-13]
> distinctFaces(19,6,2)
[7864290 609359740010496 48545/3761479876608 1.2905825 E-8 1.2905825 E-6]
> distinctFaces(19,6,3)
[23213772120 609359740010496 967240505/25389989167104 3.8095349 E-5 0.0038095349]
> distinctFaces(19,6,4)
[4053480102000 609359740010496 28149167375/4231664861184 0.0066520314 0.66520314]
> distinctFaces(19,6,5)
[106264284991200 609359740010496 368973211775/2115832430592 0.17438678 17.438678]
> distinctFaces(19,6,6)
[499018753280880 609359740010496 1155136002965/1410554953728 0.81892308 81.892308]
> distinctFaces(20,6,1)
[6 3656158440062976 1/609359740010496 1.6410667 E-15 1.6410667 E-13]
> distinctFaces(20,6,2)
[15728610 3656158440062976 2621435/609359740010496 4.3019498 E-9 4.3019498 E-7]
> distinctFaces(20,6,3)
[69672773520 3656158440062976 483838705/25389989167104 1.9056279 E-5 0.0019056279]
> distinctFaces(20,6,4)
[16283561724360 3656158440062976 75386859835/16926659444736 0.0044537353 0.44537353]
> distinctFaces(20,6,5)
[539428385160000 3656158440062976 312169204375/2115832430592 0.14753966 14.753966]
> distinctFaces(20,6,6)
[3100376804676480 3656158440062976 2691299309615/3173748645888 0.84798754 84.798754]