How many even numbers between 20000 and 40000, including the extremes, can be formed using the digits 0, 2, 3, 4, 6, and 8 if repetition is allowed?

Question

13-05-2025
Mathematics

High School

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How many even numbers between 20000 and 40000, including the extremes, can be formed using the digits 0, 2, 3, 4, 6, and 8 if repetition is allowed?

Answer :

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Final answer:

There are 1,080 even numbers that can be formed using the digits 0, 2, 3, 4, 6, and 8 between 20,000 and 40,000 with repetition allowed. This calculation considers that the tens place can attain any of the available digits, but the units place must end with an even number. No valid numbers can be formed between 30,000 and 40,000 with the given digits.

Explanation:

The question asks for the number of even numbers between 20,000 and 40,000 that can be formed using the digits 0, 2, 3, 4, 6, and 8 with repetition allowed. An even number must end in 0, 2, 4, 6, or 8. Considering the place values, in the ten-thousands place, we can only have 2 or 3 (2 options).

In the thousands, hundreds, and tens places, any of the digits 0, 2, 3, 4, 6, and 8 can be used (6 options each). For the units place to ensure the number is even, we can only have 0, 2, 4, 6, or 8 (5 options). Calculating the total combinations, we have:

1 option for the ten-thousands place (only the digit '2' fits the condition).
6 options for the thousands place.
6 options for the hundreds place.
6 options for the tens place.
5 options for the units place (to ensure the number is even).

Multiplying these together: 1 × 6 × 6 × 6 × 5 = 1,080 combinations. This is the total number of even numbers that can be formed between 20,000 and 30,000. Since we can't use the digit '4' in the ten-thousands place, we can't form any valid numbers between 30,000 and 40,000. Therefore, our final answer is 1,080.