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5. a) How many even numbers can be formed from the digits 1,2,3,4,5 ? b) How many of these numbers are greater than 3000 ?

User HenryTK
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2 Answers

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Final answer:

To determine the number of even numbers that can be formed from the digits 1, 2, 3, 4, 5, we need to consider the restrictions on the first digit. The total number of even numbers that can be formed is 250. To determine the number of these even numbers that are greater than 3000, we need to focus on the thousands digit. The total number of even numbers greater than 3000 that can be formed is 500.

Step-by-step explanation:

To determine the number of even numbers that can be formed from the digits 1, 2, 3, 4, 5, we need to consider the restrictions on the first digit. Since the number should be even, the units digit can only be 2 or 4. For the remaining digits, any of the available digits can be used, giving us 5 choices for each of the remaining three digits.

Therefore, the total number of even numbers that can be formed is 2 * 5 * 5 * 5 = 250.

To determine the number of these even numbers that are greater than 3000, we need to focus on the thousands digit. Since the number should be greater than 3000, the thousands digit can only be 3, 4, or 5. For the remaining three digits, any of the available digits can be used, giving us 5 choices for each digit.

Therefore, the total number of even numbers greater than 3000 that can be formed is 4 * 5 * 5 * 5 = 500

User Evens
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1 vote

Final answer:

There are 48 even numbers that can be formed from the digits 1, 2, 3, 4, 5. Among these, 12 even numbers are greater than 3000. The calculation considers the position of even and odd digits based on the value constraints.

Step-by-step explanation:

Even Numbers Formed from the Digits 1, 2, 3, 4, 5

To determine how many even numbers can be formed from the digits 1, 2, 3, 4, 5, we need to remember that an even number must end with an even digit. Here, the possible even digits are 2 and 4. If we fix an even digit at the unit place, we have 4 remaining places that can be filled with the remaining digits in any order.

For the unit place, we have 2 options (2 or 4). For the tens place, we have 4 remaining digits, and for each of these, we have 3 options for the hundreds place, then 2 for the thousands, and finally 1 for the ten-thousands. So the total number of even numbers is:

2 (for unit place) × 4 (for tens place) × 3 (for hundreds place) × 2 (for thousands place) × 1 (for ten-thousands place) = 48.

Even Numbers Greater Than 3000

To find even numbers greater than 3000 formed from the same digits, we need to fix the thousands place with a digit that is 3 or greater (3 or 4 since 5 would result in an odd number). We have 2 options for the thousands place (3 or 4). Suppose we use 4 for the thousands place; now we cannot use it for the units place, so we only have 1 even digit left for the units place (2). Thus, the counting is as follows:

2 (for thousands place) × 1 (for unit place, must be 2 if thousand place is 4) × 3 (remaining options for tens) × 2 (for hundreds) × 1 (for the remaining digit) = 12.

Therefore, there are 48 even numbers that can be formed from the given digits, and among these, 12 are greater than 3000.

User Anilcemsimsek
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