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Rule of Sum and Rule of Product

Seek simple and succinct solutions in these systems by sussing-out the significant symmetries.

Counting Integers in a Range

         

How many integers \(n\) satisfy the inequality \( 22 \leq n \leq 49\)?

How many integers are greater than or equal to \( 986\) but strictly less than \(1031?\)

How many integers \(n\) are there satisfying the inequality \[479 < n < 559?\]

How many 2 digit numbers are there, such that the units digit is strictly smaller than the tens digit?

Details and assumptions

The number \(12=012\) is a 2-digit number, not a 3-digit number.

Han Dynasty is the second imperial dynasty of China. It was established in 206 BCE and survived until 220 CE. How many years are there between its establishment and collapse inclusive of both ends?

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