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.
