Combination Type Formula Shortcut

EMAN wrote:There was a problem asking how many flights between a pair of 50 cities were possible.

Notice that you can also solve this problem using counting principles. It's not clear whether the order should matter (without seeing the wording of the original question); that is, do we consider a flight from LA to Chicago to be different from a flight from Chicago to LA? If so, then when choosing a flight we have 50 choices for the first city and 49 for the second, so 50*49 = 2450 flights in total. If order is not important, then we need to divide by 2, so we have (50*49)/2 = 1225 flights (this is sometimes written as 50C2, the number of ways of choosing 2 things from 50 if order does not matter).

Note also that if order is not important, this is, mathematically speaking, the same problem as "50 people at a party all shake hands. How many handshakes take place?", or "In a tennis tournament, 50 players each play exactly one match against each other opponent. How many matches are played?", or "50 distinct dots are drawn on a page. How many lines can be drawn which connect pairs of dots?" See Q121 in the PS section of OG12 for a similar example.

There are a lot of counting problems that look different on the surface, but all boil down to the same basic idea.