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How many even integers between 4000 and 7000 have four different digits?
1. 672
2. 840
3. 504
4. 728

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Correct Answer - Option 4 : 728

Concept:

Find the total number of thousand place, hundred places and unit digit.

Calculation:

In 4000 to 7000 , the thousand place digit are \(\in {4, 5,6}\)

Case 1: Thousand place digit is even, i.e., 4, 6

⇒ Then there are only \(\dfrac {4+5}{2}-1 = 4\) possibilities for the units digit.

This leaves 8 possible digits for the hundreds place and 7 for the tens places, 

⇒ Total of even integers = 2. 8.7.4 = 448


Case 2: Thousand place digit is odd, i.e., 5

⇒ There are 5 choices for the units digit, with 8 digits for the hundreds and 7 for the tens place. This gives 1.8.7.5 = 280 possibilities.

The total number of even integers = 448 + 280 = 728

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