9. Alison has a set of ten fridge magnets showing the integers from 0 to 9 inclusive.
In how many different ways can she split the set into five pairs so that the sum of each pair is a multiple of 5 ?
A 1
B 2
C 3
D 4
E 5
SolUTION
D

The number 0 can only be paired with 5 .
The number 1 may be paired with 4 or with 9 . If 1 is paired with 4,6 has to be paired with 9 . If 1 is paired with 9,6 has to be paired with 4 .
The number 2 may be paired with 3 or with 8 . If 2 is paired with 3,7 has to be paired with 8 . If 2 is paired with 8,7 has to be paired with 3 .
These possibilities are shown in the diagram above. Thus the complete pairing is determined by first the choice which of 4 or 9 to pair with 1, giving two choices, and then the choice of which of 3 or 8 to pair with 2 .
These choices are independent. It follows that there are $2 \times 2=4$ ways to split the set of numbers into five pairs so that the sum of each pair is a multiple of 5 .