Ten children are to be divided into an a team and a b team of 5 each. the a team will play in one league and the b team in another. how many different divisions are possible?

The number of combinations for picking a number of items from a given selection is found using this formula: nCr = n!/(n - r)!r! where:
n = size of the selection
r = number of items being picked
In the case of the question, there is a selection of 10 people of which 5 will be chosen to be in team a so:
n = 10
r = 5
The number of combinations is 10C5 = 252
This means there are 252 possible different a teams that could be made from the 10 people.
Team b will be another of the 252 combinations.
Note: scientific calculators, e.g. Casio ones, have a button for the nCr formula.