Question 1 of 11

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Let the format of digital clock be $h:m_1m_2:s_1s_2$. For h there is only one choice, namely 7. $m_1$ and $s_1$ each to be filled in with digits 0 to 5, 6 choices for $m_1$ and 5 choices for $s_1$ $($different from $m_1)$. $m_2$ and $s_2$ each to be filled in with digits 0 to 9, but not 7 and not the numbers used in $m_1$ and $s_1$, so 7 choices for $m_2$ and 6 choices for $s_2$ $($different from $m_2)$. \begin{equation*} 1\times 6 \times 7 \times 5 \times 6 = 1260 \end{equation*}

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Let the format of digital clock be $h:m_1m_2:s_1s_2$. For h there is only one choice, namely 0. $m_1$ and $s_1$ each to be filled in with digits 0 to 5, 5 choices for $m_1$ and 4 choices for $s_1$ (different from $m_1$). $m_2$ and $s_2$ each to be filled in with digits 0 to 9, but not 0 and not the numbers used in $m_1$ and $s_1$, so 7 choices for $m_2$ and 6 choices for $s_2$ $($different from $m_2)$. \begin{equation*} 1 \times 5 \times 7 \times 4 \times 6 = 840 \end{equation*}

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Let the format of digital clock be $h:m_1m_2:s_1s_2$. For h there is only one choice, namely 1. $m_1$ and $s_1$ each to be filled in with digits 0 to 5, 5 choices for $m_1$ and 4 choices for $s_1$ (different from $m_1$). $m_2$ and $s_2$ each to be filled in with digits 0 to 9, but not 1 and not the numbers used in $m_1$ and $s_1$, so 7 choices for $m_2$ and 6 choices for $s_2$ $($different from $m_2)$. \begin{equation*} 1 \times 5 \times 7 \times 4 \times 6 = 840 \end{equation*}

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Let the format of digital clock be $h:m_1m_2:s_1s_2$. For h there is only one choice, namely 2. $m_1$ and $s_1$ each to be filled in with digits 0 to 5, 5 choices for $m_1$ and 4 choices for $s_1$ (different from $m_1$). $m_2$ and $s_2$ each to be filled in with digits 0 to 9, but not 2 and not the numbers used in $m_1$ and $s_1$, so 7 choices for $m_2$ and 6 choices for $s_2$ $($different from $m_2)$. \begin{equation*} 1 \times 5 \times 7 \times 4 \times 6 = 840 \end{equation*}

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Let the format of digital clock be $h:m_1m_2:s_1s_2$. For h there is only one choice, namely 3. $m_1$ and $s_1$ each to be filled in with digits 0 to 5, 5 choices for $m_1$ and 4 choices for $s_1$ (different from $m_1$). $m_2$ and $s_2$ each to be filled in with digits 0 to 9, but not 3 and not the numbers used in $m_1$ and $s_1$, so 7 choices for $m_2$ and 6 choices for $s_2$ $($different from $m_2)$. \begin{equation*} 1 \times 5 \times 7 \times 4 \times 6 = 840 \end{equation*}

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Let the format of digital clock be $h:m_1m_2:s_1s_2$. For h there is only one choice, namely 4. $m_1$ and $s_1$ each to be filled in with digits 0 to 5, 5 choices for $m_1$ and 4 choices for $s_1$ (different from $m_1$). $m_2$ and $s_2$ each to be filled in with digits 0 to 9, but not 4 and not the numbers used in $m_1$ and $s_1$, so 7 choices for $m_2$ and 6 choices for $s_2$ $($different from $m_2)$. \begin{equation*} 1 \times 5 \times 7 \times 4 \times 6 = 840 \end{equation*}

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Let the format of digital clock be $h:m_1m_2:s_1s_2$. For h there is only one choice, namely 5. $m_1$ and $s_1$ each to be filled in with digits 0 to 5, 5 choices for $m_1$ and 4 choices for $s_1$ (different from $m_1$). $m_2$ and $s_2$ each to be filled in with digits 0 to 9, but not 5 and not the numbers used in $m_1$ and $s_1$, so 7 choices for $m_2$ and 6 choices for $s_2$ $($different from $m_2)$. \begin{equation*} 1 \times 5 \times 7 \times 4 \times 6 = 840 \end{equation*}

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Let the format of digital clock be $h:m_1m_2:s_1s_2$. For h there is only one choice, namely 6. $m_1$ and $s_1$ each to be filled in with digits 0 to 5, 6 choices for $m_1$ and 5 choices for $s_1$ (different from $m_1$). $m_2$ and $s_2$ each to be filled in with digits 0 to 9, but not 6 and not the numbers used in $m_1$ and $s_1$, so 7 choices for $m_2$ and 6 choices for $s_2$ $($different from $m_2)$. \begin{equation*} 1 \times 6 \times 7 \times 5 \times 6 = 1260 \end{equation*}

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Let the format of digital clock be $h:m_1m_2:s_1s_2$. For h there is only one choice, namely 7. $m_1$ and $s_1$ each to be filled in with digits 0 to 5, 6 choices for $m_1$ and 5 choices for $s_1$ (different from $m_1$). $m_2$ and $s_2$ each to be filled in with digits 0 to 9, but not 7 and not the numbers used in $m_1$ and $s_1$, so 7 choices for $m_2$ and 6 choices for $s_2$ $($different from $m_2)$. \begin{equation*} 1 \times 6 \times 7 \times 5 \times 6 = 1260 \end{equation*}

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Let the format of digital clock be $h:m_1m_2:s_1s_2$. For h there is only one choice, namely 8. $m_1$ and $s_1$ each to be filled in with digits 0 to 5, 6 choices for $m_1$ and 5 choices for $s_1$ (different from $m_1$). $m_2$ and $s_2$ each to be filled in with digits 0 to 9, but not 8 and not the numbers used in $m_1$ and $s_1$, so 7 choices for $m_2$ and 6 choices for $s_2$ $($different from $m_2)$. \begin{equation*} 1 \times 6 \times 7 \times 5 \times 6 = 1260 \end{equation*}

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Let the format of digital clock be $h:m_1m_2:s_1s_2$. For h there is only one choice, namely 9. $m_1$ and $s_1$ each to be filled in with digits 0 to 5, 6 choices for $m_1$ and 5 choices for $s_1$ (different from $m_1$). $m_2$ and $s_2$ each to be filled in with digits 0 to 9, but not 9 and not the numbers used in $m_1$ and $s_1$, so 7 choices for $m_2$ and 6 choices for $s_2$ $($different from $m_2)$. \begin{equation*} 1 \times 6 \times 7 \times 5 \times 6 = 1260 \end{equation*}