Probability of the maximum of n binomial random variables being less than n
Solution 1:
$$\mathbb{P}[X_i<n]=1-\mathbb{P}[X_i=n]=(1-p^n)$$
Thus
$$\mathbb{P}[\max\{X_1,\dots,X_n\}<n]=\mathbb{P}[X_1<n]^n=(1-p^n)^n$$
Probability of the maximum of n binomial random variables being less than n
Solution 1:
$$\mathbb{P}[X_i<n]=1-\mathbb{P}[X_i=n]=(1-p^n)$$
Thus
$$\mathbb{P}[\max\{X_1,\dots,X_n\}<n]=\mathbb{P}[X_1<n]^n=(1-p^n)^n$$