Shouldn't I Post This In July?

$CUBAJULY1987\quad can\quad be\quad obtained\quad on\quad the\quad 1st\quad to\quad 12th\quad draws,\\ 2nd\quad to\quad 13th\quad draws,\quad 3rd\quad to\quad 14th\quad draws,\quad ...,\quad and\quad 17th\quad to\quad 28th\\ draws.\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ The\quad number\quad of\quad CUBAJULY1987\quad that\quad can\quad appear\quad in\quad the\quad sequence\\ is\quad at\quad most\quad 2.\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ Assuming\quad that\quad the\quad 2\quad sets\quad of\quad CUBAJULY1987,\quad if\quad there\quad are\quad 2\quad sets\quad in\quad the\quad sequence,\\ are\quad distinct,\quad there\quad are\quad 17\times { 20 }^{ 16 }\quad ways\quad such\quad that\quad there\quad is\quad at\quad least\\ one\quad set\quad of\quad CUBAJULY1987\quad present\quad in\quad the\quad sequence.\quad (17\quad from\quad the\\ 17\quad "spots"\quad that\quad the\quad 1st\quad CUBAJULY1987\quad can\quad occupy\quad and\quad { 20 }^{ 16 }\quad for\\ the\quad remaining\quad "spots")\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ However,\quad in\quad the\quad 17\times { 20 }^{ 16 }\quad such\quad ways,\quad each\quad case\quad where\quad there\quad are\quad 2\quad sets\\ of\quad CUBAJULY1987\quad is\quad counted\quad twice\quad and\quad so\quad the\quad total\quad number\quad of\quad cases\\ where\quad there\quad are\quad 2\quad sets\quad of\quad CUBAJULY1987\quad must\quad be\quad subtracted\quad to\quad the\\ 17\times { 20 }^{ 16 }\quad ways\quad to\quad get\quad the\quad total\quad number\quad of\quad ways\quad such\quad that\quad CUBAJULY1987\\ appears\quad in\quad the\quad sequence.\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ The\quad total\quad number\quad of\quad cases\quad where\quad there\quad are\quad 2\quad sets\quad of\quad CUBAJULY1987\quad present\\ is\quad 15\times { 20 }^{ 4 }.\quad (15\quad from\quad the\quad total\quad number\quad of\quad "spots"\quad the\quad 2\quad sets\quad can\quad occupy\quad and\quad \\ { 20 }^{ 4 }\quad for\quad the\quad remaining\quad "spots")\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ So,\quad the\quad total\quad number\quad of\quad ways\quad such\quad that\quad CUBAJULY1987\quad occurs\quad in\quad the\quad sequence\quad is\\ 17\times { 20 }^{ 16 }-15\times { 20 }^{ 4 }.\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ The\quad total\quad number\quad of\quad sequences\quad possible\quad is\quad { 20 }^{ 28 }.\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ Thus,\quad the\quad probability\quad that\quad CUBAJULY1987\quad occurs\quad in\quad the\quad sequence\quad is\quad \frac { 17\times { 20 }^{ 16 }-15\times { 20 }^{ 4 } }{ { 20 }^{ 28 } } .\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ Therefore,\quad a+b+c+d+e+f=17+20+16+15+4+28=100.$