From a deck of 52 cards, the 12 face cards and 4 aces are removed. From these 16 cards that are removed, 4 are chosen. How many combinations are possible that have at most 1 red card?

Of the 16 cards selected, you have 8 red and 8 black.
So selection of atmost 1 red card is, Either the selection has no red or one red.
i) Selection of no red = All black = C(8,4) = 8!/(4! x 4!) = 70
ii) Selection of one red = Selection of one red x Selection of 3 black
= C(8,1) x C(8,3) = 8 x 56 = 448
Hence total number of combinations possible = 70 + 448 = 518
So selection of atmost 1 red card is, Either the selection has no red or one red.
i) Selection of no red = All black = C(8,4) = 8!/(4! x 4!) = 70
ii) Selection of one red = Selection of one red x Selection of 3 black
= C(8,1) x C(8,3) = 8 x 56 = 448
Hence total number of combinations possible = 70 + 448 = 518