Seven cards are marked with the numbers 1, 2,3, 4, 5, 6, and 7 , then shuffled, and two cards are drawn. how many different 2 -card combinations are possible?

There are 7 choices for the first card, 6 choices for the second, so 7*6=42 ways of choosing 2 cards IF order counts.

Since 2-card combinations do not care about order, we have over-counted the combinations by a factor of two (e. g. AB is the same combination as BA).

So the number of combinations is 7*6/2=21.

Mathematically, the formula to use is for choosing r-card combinations from n cards is

C (n, r) = n! / (r! (n-r) !) = 7! / (2!5!) = 7*6 / (2*1) = 21 as before.