Answer
The answer is approximately 0.847987541
Solution
The answer could be approximated as expressed 1 - (prob(no 1's) + prob(no 2's) + ... + prob(no 6's)) = 1 - 6*(5/6)^20 = apx. 0.84349568.
However that would double-subtract the situations where two different sides never got rolled. There are combin(6,2)=15 ways to choose two sides out of six. The probability that any two given sides never get rolled is (4/6)^20. We need to add those to the probability, because they got subtracted twice in the previous step. So, now we're at 1 - 6*(5/6)^20 + 15*(4/6)^20 = apx. 0.84800661.
This question is asked and discussed in my forum at Wizard of Vegas.
However, if any group of three sides that had never been rolled would have been triple-subtracted in the first step and triple-added in the second step. We need to subtract them back out as a state where not all six sides were rolled. There are combin(6,3) = 20 ways to choose three sides out of six. The probability that any specific three sides are never rolled is (3/6)^20. So, now we're at 1 - 6*(5/6)^20 + 15*(4/6)^20 - 20*(3/6)^20= apx. 0.847987537.
However, if any group of four sides that had never been rolled would have been quadruple-subtracted in the first step, quadruple-added in the second step, and quadruple subtracted in the third step. We need to add them back in, because each such state was already subtracted out twice. There are combin(6,4) = 15 ways to choose four sides out of six. The probability that any specific four sides are never rolled is (2/6)^20. So, now we're at 1 - 6*(5/6)^20 + 15*(4/6)^20 - 20*(3/6)^20 + 15*(2/6)^20 = apx. 0.84798754089.
However, if all 20 rolls were the same numbers, this situation would have been quintuple-subtracted in the first step, quintuple-added in the first step, quintuple-subtracted in the third step, and quintuple-added in the fourth step. We need to subtracted them back out. So, now we're at 1 - 6*(5/6)^20 + 15*(4/6)^20 - 20*(3/6)^20 + 15*(2/6)^20 - 6*(1/6)^20 = apx. 0.84798754089.
So the answer is 1-6*(5/6)^20+COMBIN(6,4)*(4/6)^20-COMBIN(6,3)*(3/6)^20+COMBIN(6,2)*(2/6)^20-6*(1/6)^20 = apx. 0.84798754089.