In a $12$-hour clock that runs correctly, how many times do the second, minute, and hour hands of the clock coincide, in a $12$-hour duration from $3$ PM in a day to $3$ AM the next day?

1. $11$
2. $12$
3. $144$
4. $2$

The hands of a clock coincide $11$ times in every $12$ hour (Since between $11$ and $1,$ they coincide only once, i.e., at $12$ o'clock). The hands overlap about every $65$ minutes, not every $60$ minutes.

We can see in this way:

${\color{Green}{\text{3 PM}}} \overset{1}{\longrightarrow} {\color{Blue}{\text{4:05 PM}}} \overset{2}{\longrightarrow} {\color{Teal}{\text{5:10 PM}}} \overset{3}{\longrightarrow} {\color{Purple}{\text{6:15 PM}}} \overset{4}{\longrightarrow} {\color{Lime}{\text{7:20 PM}}} \overset{5}{\longrightarrow} {\color{Violet}{\text{8:25 PM}}} \overset{6}{\longrightarrow} {\color{Cyan}{\text{9:30 PM}}} \overset{7}{\longrightarrow} {\color{Olive}{\text{10:35 PM}}} \overset{8}{\longrightarrow} {\color{Magenta}{\text{11:40 PM}}} \overset{9}{\longrightarrow} {\color{Orange}{\text{12:45 AM}}} \overset{10}{\longrightarrow} {\color{DarkOrchid}{\text{1:50 AM}}} \overset{11}{\longrightarrow} {\color{Red}{\text{2:55 AM}}}$

Correct Answer $:\text{A}$
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