Probability of the maximum of n binomial random variables being less than n

Question

Suppose $x_1$, $x_2$, ..., $x_n$ are independent binomial random variables with parameters $p$ and $n$. Now what is the probability of $P[\text{max}(x_{1}, x_{2}, ..., x_{n})<n]$? I think we can simplify the question to each $x_i$ being less than $n$, i.e. $(P[x_{i}<n])^n$. then, $(\binom{n}{x}p^{x}(1-p)^{n-x})^{n}<n$. But I do not know how to proceed.

Answer 1

$$\mathbb{P}[X_i<n]=1-\mathbb{P}[X_i=n]=(1-p^n)$$

Thus

$$\mathbb{P}[\max\{X_1,\dots,X_n\}<n]=\mathbb{P}[X_1<n]^n=(1-p^n)^n$$