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Is it true that that ${{n}\choose{k}} p^k (1-p)^{n-k}$ is dominated by $\frac{1}{n}$, at least for $k$ sufficiently big?

EDIT: I saw that the question was absolutely not stated as I intended. The real question is: given a real number $p \in (0, 1)$, is there some $\bar{k}$ big enough such that for every $k \geq \bar{k}$, for every $n \geq k$, ${{n}\choose{k}} p^k (1-p)^{n-k} \leq \frac{C}{n},$ for some constant $C$?

Is it true that that ${{n}\choose{k}} p^k (1-p)^{n-k}$ is dominated by $\frac{1}{n}$, at least for $k$ sufficiently big?

EDIT: I saw that the question was absolutely not stated as I intended. The real question is: given a real number $p \in (0, 1)$, assuming $k$ fixed big enough, is there some constant $C > 0$ independent from $k$ such that $n \cdot {{n}\choose{k}} q^k (1-q)^{n-k} \leq C$ for every $n \geq k$?

Is it true that that ${{n}\choose{k}} p^k (1-p)^{n-k}$ is dominated by $\frac{1}{n}$, at least for $k$ sufficiently big?

EDIT: I saw that the question was absolutely not stated as I intended. The real question is: given a real number $p \in (0, 1)$, is there some $\bar{k}$ big enough such that for every $k \geq \bar{k}$, for every $n \geq k$, ${{n}\choose{k}} p^k (1-p)^{n-k} \leq \frac{C}{n}, \ \forall n \geq k$ for some constant $C$?

Is it true that that ${{n}\choose{k}} p^k (1-p)^{n-k}$ is dominated by $\frac{1}{n}$, at least for $k$ sufficiently big?

EDIT: I saw that the question was absolutely not stated as I intended. The real question is: given a real number $p \in (0, 1)$, is there some $\bar{k}$ big enough such that for every $k \geq \bar{k}$, for every $n \geq k$, ${{n}\choose{k}} p^k (1-p)^{n-k} \leq \frac{C}{n},$ for some constant $C$?

Is it true that that ${{n}\choose{k}} p^k (1-p)^{n-k}$ is dominated by $\frac{1}{n}$, at least for $k$ sufficiently big?

Is it true that that ${{n}\choose{k}} p^k (1-p)^{n-k}$ is dominated by $\frac{1}{n}$, at least for $k$ sufficiently big?

EDIT: I saw that the question was absolutely not stated as I intended. The real question is: given a real number $p \in (0, 1)$, is there some $\bar{k}$ big enough such that for every $k \geq \bar{k}$, for every $n \geq k$, ${{n}\choose{k}} p^k (1-p)^{n-k} \leq \frac{C}{n}, \ \forall n \geq k$ for some constant $C$?