I guess your equation is wrong, with c=1 the fraction is always reduced to "0", although it should be d/30 for d cards drawed.
I hope I am not wrong with this statement, maybe I have something on my eyes and its already late so be dont be angry with me if I am wrong.
My equation:
6 copies , 7 draws and 30 total to get AT LEAST 1:
p(X>0)=1-p(X=0)
=1 - 24/30 * 23/29* 22/28 * 21/27 * 20/26 * 19/25 * 18/24
the upper sequence: 24, 23, 22, 21, 20, 19, 18 is done with: (t-c)! / (t-c-d)! --> (30-6)! / (30-6-7)! --> (24)! / (17)! --> 24*23*22*21*20*19*18
the lower sequence: 30, 29, 28, 27, 26 ,25, 24 is done with: (t)! / (t-d)! --> (30)! / (30-7)! --> (30)! / (23)! --> 30*29*28*27*26*25*24
Your lower sequence will be 29*28*27*26*25*24*23, because the "-1".
I cant imagine that the "-1" will realise "exactly 1", but I will think about that tomorrow or the day after. Maybe you find my or your fault.