Binomial Trouble!

Consider the alternating sum of binomial coefficients below: $\binom n1 - \binom n3 + \binom n5 - \binom n7 + \cdots \pm \binom nm = 0,$ where $m$ denotes the largest odd number less than or equal to $n$ .

Which of the following options must be true for any positive integer $k$ ? Justify your answer.

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Notation: $\dbinom MN = \dfrac {M!}{N! (M-N)!}$ denotes the binomial coefficient .