For our next problem, let us calculate the probabilty of getting all 20 numbers after rolling a 20 sided die 67 times. (Incidentally, a 20 sided die is called an icosahedron and they are used in many games and toys. For example, the "Magic 8Ball^{™}" contains an icosahedron with 20 different "answers" written on each of its 20 faces.)
STEP 1
which equals 1.48 × 10^{87}
Then in steps 2, 3, 4 and 5, we will determine how many of those 20^{67} rolls, will contain all 20 numbers.
STEP 2 19^{67} = 4.75 × 10^{85} 18^{67} = 1.27 × 10^{84} 17^{67} = 2.75 × 10^{82} 16^{67} = 4.74 × 10^{80} 15^{67} = 6.28 × 10^{78} 14^{67} = 6.17 × 10^{76} 13^{67} = 4.31 × 10^{74} 12^{67} = 2.02 × 10^{72} 11^{67} = 5.93 × 10^{69} 10^{67} = 1.00 × 10^{67} 9^{67} = 8.60 × 10^{63} 8^{67} = 3.21 × 10^{60} 7^{67} = 4.18 × 10^{56} 6^{67} = 1.37 × 10^{52} 5^{67} = 6.78 × 10^{46} 4^{67} = 2.18 × 10^{40} 3^{67} = 9.27 × 10^{31} 2^{67} = 1.48 × 10^{20} 1^{67} = 1
STEP 3 19 C 20 = 20 18 C 20 = 190 17 C 20 = 1,140 16 C 20 = 4,845 15 C 20 = 15,504 14 C 20 = 38,760 13 C 20 = 77,520 12 C 20 = 125,970 11 C 20 = 167,960 10 C 20 = 184,756 9 C 20 = 167,960 8 C 20 = 125,970 7 C 20 = 77,520 6 C 20 = 38,760 5 C 20 = 15,504 4 C 20 = 4,845 3 C 20 = 1,140 2 C 20 = 190 1 C 20 = 20 Basically, this is saying that 19 objects can be chosen from a set of 20 in 20 ways 18 objects can be chosen from a set of 20 in 190 ways .......................................................................................
2 objects can be chosen from a set of 20 in 190 ways
STEP 4 4.75E+085 × 20 = 9.50 × 10^{86} 1.27E+084 × 190 = 2.41 × 10^{86} 2.75E+082 × 1,140 = 3.14 × 10^{85} 4.74E+080 × 4,845 = 2.30 × 10^{84} 6.28E+078 × 15,504 = 9.74 × 10^{82} 6.17E+076 × 38,760 = 2.39 × 10^{81} 4.31E+074 × 77,520 = 3.34 × 10^{79} 2.02E+072 × 125,970 = 2.54 × 10^{77} 5.93E+069 × 167,960 = 9.97 × 10^{74} 1.00E+067 × 184,756 = 1.85 × 10^{72} 8.60E+063 × 167,960 = 1.44 × 10^{69} 3.21E+060 × 125,970 = 4.05 × 10^{65} 4.18E+056 × 77,520 = 3.24 × 10^{61} 1.37E+052 × 38,760 = 5.30 × 10^{56} 6.78E+046 × 15,504 = 1.05 × 10^{51} 2.18E+040 × 4,845 = 1.06 × 10^{44} 9.27E+031 × 1,140 = 1.06 × 10^{35} 1.48E+020 × 190 = 2.80 × 10^{22} 1 × 20 = 20
STEP 5
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