Bali Pairs Not able to understand the logic

Just not able to understand the question.Moreover, hint is also not helpful.Please help on this.

@Vishal_234 in this problem you have been given N pair of shoes from each pair you have to select one shoe means at end you should have N shoes.so from all the possible combination of selecting one shoe from each pair given you have N pair count the number of combinations in which adding the selected shoe sizes gives you an odd value atlast.
-try thinking a dp approach.

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