A box contains seven cards, each with a different integer from $$1$$ to $$7$$ written on it. Avani takes three cards from the box and then Niamh takes two cards, leaving two cards in the box. Avani looks at her cards and then tells Niamh "I know the sum of the numbers on your cards is even." What is the sum of the numbers on Avani\textquotesingle s cards? 
Overseas Competition->Knowledge Point->Counting Modules->Permutations and Combinations->Combinations
To obtain an even number when adding two integers, both integers must be even or both integers must be odd. Therefore the four integers remaining once Avani has removed her three integers must all be odd or all be even or there would be a possibility that the sum of Niamh\textquotesingle s two integers could be odd. Since there were four odd integers and three even integers on the cards in the box initially, the integers on the cards remaining once Avani has removed her cards are all odd. Therefore the cards Avani removed had the three even integers $$2$$, $$4$$ and $$6$$ written on them which have sum $$12$$. 