Review question

# Which dates contain no repetitions of a digit? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R5212

## Question

This question concerns calendar dates of the form $d_1d_2/m_1m_2/y_1y_2y_3y_4$ in the order day/month/year.

The question specifically concerns those dates which contain no repetitions of a digit. For example, the date $23/05/1967$ is such a date but $07/12/1974$ is not such a date as both $1=m_1=y_1$ and $7=d_2=y_3$ are repeated digits.

We will use the Gregorian Calendar throughout (this is the calendar system that is standard throughout most of the world; see below.)

1. Show that there is no date with no repetition of digits in the years from $2000$ to $2099$.
2. What was the last date before today, $03/11/2010$, with no repetition of digits? Explain your answer.
3. When will the next such date be? Explain your answer.
4. How many such dates were there in years from $1900$ to $1999$? Explain your answer.

[The Gregorian Calendar uses 12 months, which have, respectively, $31$, $28$ or $29$, $31$, $30, 31, 30, 31, 31, 30, 31, 30$ and $31$ days. The second month (February) has $28$ days in years that are not divisible by $4$, or that are divisible by $100$ but not $400$ (such as $1900$); it has $29$ days in the other years (leap years).]