<s>[INST] We perform $n$ independent trials, where each trial has two equally likely outcomes: success or failure. Let $X$ be the random variable that represents the number of failures in the $n$ trials, where I assume that $X$ takes the values $0, 1, \ldots, n$. We now consider three events: Event $A$: $X \le 10$; Event $B$: $n - X \ge 7$; and Event $C$: $n - X = 6$. Show that $A \cup B \cup C = X \le 10$ and $A \cap B \cap C = \emptyset$.[/INST]