2. In how many different ways can we exchange (break down) a 50 euro banknote using 2 euro coins and 5 and 10 euro banknotes? Write all the possibilities.
First solution.

To have the total value of all selected banknotes be 50, among them:

a 10-euro banknote can be $0,1,2,3,4$ or 5,

a 5-euro banknote can be $0,2,4,6,8$ or 10,

and a 2-euro coin can be $0,5,10,15,20$ or 25.

2 POINTS

We list the triples, first aligning the 2-euro coins, then the 5-euro banknotes, and finally the 10-euro banknotes:

$(0,0,5),(0,2,4),(0,4,3),(0,6,2),(0,8,1),(0,10,0)$

$(5,0,4),(5,2,3),(5,4,2),(5,6,1),(5,8,0)$

$(10,0,3),(10,2,2),(10,4,1),(10,6,0)$

$(15,0,2),(15,2,1),(15,4,0)$

$(20,0,1),(20,2,0)$

$(25,0,0)$

each set of three triples earns 1 POINT

There are a total of 21 different ways.

1 POINT

Note: If a student lists all the triples and answers the question (1 POINT) without the introductory explanation, which is worth 2 POINTS, they receive a total of 10 POINTS. If they systematically list the triples but miss one solution, they receive a total of 8 POINTS, and if they miss 2 or 3 solutions, they receive a total of 7 POINTS.

TOTAL 10 POINTS

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