Question

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,  4 employees in Edinburgh and 3 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?

Answer 1

1. How many teams may be chosen altogether?

total employees = 10 + 4 + 3 = 17

to be chosen = 5

so no. of possible combinations

=

answer is 6188

(2) all the members of the Aberdeen office?

so 3 out of 5 have already been decided.

so possible choices = 10 + 4 = 14

to be chosen = 5 - 3 = 2

so combinations

=

answer is 91

(3) no members from the Edinburgh office?

so possible choices = 10 + 3 = 13

to be chosen = 5

so combinations

=

answer is 1287

(4) at least one members from the Glasgow office?

case 1 : exactly one from Glasgow

so possible choices = 10 + 4 = 14

to be chosen = 4

so combinations = ¹⁴C₄ = 1001

case 2 : exactly 2 from Glasgow

so possible choices = 10 + 4 = 14

to be chosen = 3

so combinations = ¹⁴C₃ = 364

case 3 : exactly 3 from Glasgow

so possible choices = 10 + 4 = 14

to be chosen = 5-3 = 2

so combinations = ¹⁴C₂ = 91

so total = 91 + 364 + 1001 = 1456

answer is 1456