We consider 3 type of card a, b, c and a deck with Na + Nb + Nc = 60 copies of each card.

The probably of having exactly A + B + C = 7 in a hand of 7 is given by

P(A, B, C; 60) = comb(Na, A) * comb(Nb, B) * comb(Nc, C) / comb(60, 7)

By introducing comb(60 - Na, 7 - A) in the middle we obtain

P(A, B, C; 60) = [comb(Na, A) * comb(60 - Na, 7 - A) / comb(60, 7)] * [comb(Nb, B) * comb(Nc, C) / comb(60 - Na, 7 - A)]

We recognize the probability for 2 type of card and use C = 7 - A - B to obtain

P(A, B, 7 - A - B; 60) = P(A, 7 - A; 60) * P(B, 7 - A - B, 60 - A)

Wikipedia for more detail.

The interpretation is: For each new type of card you have to remove the all the copy of the previous type of card.

This is not easy, I had to redo the proof to convince myself there was an error in the spreadsheet.