Here is an alternative solution to this problem:
Expand the generating function
(1/6x+1/6x^2+1/6x^3 + 1/6x^4+1/6x^5+1/6x^6)^10
(by using Wolfram Alpha, for example)
and find the x^20 term, which is 21,307/15,116,544=85,228/60,466,176 in this case. Each coefficient gives the probability of getting a sum of that power of x.
EXPANDED FORM:
x^60/60466176+(5 x^59)/30233088+(55 x^58)/60466176+(55 x^57)/15116544+(715 x^56)/60466176+(1001 x^55)/30233088+(185 x^54)/2239488+(35 x^53)/186624+(55 x^52)/139968+(11605 x^51)/15116544+(21307 x^50)/15116544+(36985 x^49)/15116544+(243925 x^48)/60466176+(191735 x^47)/30233088+(576565 x^46)/60466176+(23089 x^45)/1679616+(63965 x^44)/3359232+(2665 x^43)/104976+(986315 x^42)/30233088+(611575 x^41)/15116544+(2930455 x^40)/60466176+(1696805 x^39)/30233088+(3801535 x^38)/60466176+(1030315 x^37)/15116544+(240295 x^36)/3359232+(7631 x^35)/104976+(240295 x^34)/3359232+(1030315 x^33)/15116544+(3801535 x^32)/60466176+(1696805 x^31)/30233088+(2930455 x^30)/60466176+(611575 x^29)/15116544+(986315 x^28)/30233088+(2665 x^27)/104976+(63965 x^26)/3359232+(23089 x^25)/1679616+(576565 x^24)/60466176+(191735 x^23)/30233088+(243925 x^22)/60466176+(36985 x^21)/15116544+(21307 x^20)/15116544+(11605 x^19)/15116544+(55 x^18)/139968+(35 x^17)/186624+(185 x^16)/2239488+(1001 x^15)/30233088+(715 x^14)/60466176+(55 x^13)/15116544+(55 x^12)/60466176+(5 x^11)/30233088+x^10/60466176
