Counting Methods Problem { Combinations } A company has its Scottish offices in Glasgow, Edinburgh and Aberdeen and it wants to put a team of five employees together to send to negotiate new contracts abroad. There are 10 employees in the Glasgow office, 8 employees in Edinburgh and 2 employees in Aberdeen. How many teams may be chosen altogether? How many possible teams include: all the members of the Aberdeen office? no members from the Edinburgh office? *at least one member from the Glasgow office?* The last part of this questions is where i encounter difficulty, currently my work entails: 19C4 = 3876 There are 20 total possibilities and 1 has to be from Glasgow hence the 19, however i feel that i have taken a misdirection and this method is invalid. Thank You for any help provided

In this case it might be easier to use the negation of the event, in order to avoid redundancies.

**Hint:** In how many ways can you make a team with **no** members from the Glasgow office?