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help counting

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On a digital clock showing hours and minutes, how many different readings between noon and midnight contain at least one 3?

Jun 11, 2022

#1
+1746
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Lets count them case by case:

$$3: \text {__}$$ -  60 cases, 1 for each minute

$$\text _ : 3 \text _$$  - 11 cases for the hour hand (we already counted for 3), 10 cases for the ones digit, so 110 cases

$$\text{_} : \text{_} 3$$  - 11 cases for the hour hand (we already counted for 3), 6 cases for the tens digit( anything other 3), so 66 cases

In total, there are $$60 + 110 + 66 = \color{brown}\boxed{236}$$

Jun 11, 2022