Of the groups of three houses that he could select, how many are all located in the same city?

Question

Tim is a real estate investor and is going to buy 3 houses. There are 12 houses he is
looking at: 4 of them in Miami, 5 in New York, and 3 in Seattle. Of the groups of three
houses that he could select, how many are all located in the same city?

Answer 1

So, we have to figure out how many potential groups there are of all the houses being in Miami, all the houses being in New York, and all the houses being in Seattle separately (and then add them together). For all of them being in Miami, the first house chosen being in Miami could be four different houses, the second could be three different houses, and the third could be two (because there is one less house to choose from each time we pick a house). Thus the total number of combinations (not permutations because the order of the houses chosen doesn't matter) of all three houses being in Miami is 4x3x2=24. Following similarly for New York, the total number would be 5x4x3=60, and for Seattle, 3x2x1=6. Adding all of them together, 24+60+6=90. This is the number of groups of three houses Tim could select all being located in the same city.