A 4-digit number is selected from the numbers {1,2,3,4,5,6} where the digits are selected without replacement.

How many 4-digit numbers can be selected that are greater than 3000?

How many even 4-digit numbers can be chosen?

How many 4-digit numbers can be chosen that are even and greater than 3000?

To find the number of 4-digit numbers that can be selected from the numbers {1,2,3,4,5,6} where the digits are selected without replacement and are greater than 3000, we need to consider the following:

Step 1: Determine the possibilities for the thousands place digit (the leftmost digit).

Since we want the number to be greater than 3000, the thousands place digit cannot be 1 or 2. Therefore, we have 4 possibilities for the thousands place digit: 3, 4, 5, or 6.

Step 2: Determine the possibilities for the hundreds, tens, and units place digits (the remaining three digits).

For each of the thousands place digit possibilities, we have 5 remaining digits to choose from (since the digits are selected without replacement). Therefore, we have 5 possibilities for the hundreds place digit, 4 possibilities for the tens place digit, and 3 possibilities for the units place digit.

Step 3: Calculate the total number of possibilities.

To find the total number of possibilities, we multiply the number of possibilities for each digit place together.

Total number of 4-digit numbers greater than 3000 = 4 (possibilities for the thousands place digit) * 5 (possibilities for the hundreds place digit) * 4 (possibilities for the tens place digit) * 3 (possibilities for the units place digit)

Therefore, the total number of 4-digit numbers that can be selected, which are greater than 3000, is 4*5*4*3 = 240.

To find the number of 4-digit numbers that can be chosen which are also even, we need to consider the following:

Step 1: Determine the possibilities for the thousands place digit (the leftmost digit).

Since we want the number to be even, the thousands place digit cannot be 1 or 3. Therefore, we have 4 possibilities for the thousands place digit: 2, 4, 5, or 6.

Step 2: Determine the possibilities for the hundreds, tens, and units place digits (the remaining three digits).

For each of the thousands place digit possibilities, we have 5 remaining digits to choose from (since the digits are selected without replacement). Therefore, we have 5 possibilities for the hundreds place digit, 4 possibilities for the tens place digit, and 3 possibilities for the units place digit.

Step 3: Calculate the total number of possibilities.

To find the total number of possibilities, we multiply the number of possibilities for each digit place together.

Total number of even 4-digit numbers = 4 (possibilities for the thousands place digit) * 5 (possibilities for the hundreds place digit) * 4 (possibilities for the tens place digit) * 3 (possibilities for the units place digit)

Therefore, the total number of 4-digit numbers that can be chosen, which are even, is 4*5*4*3 = 240.

To find the number of 4-digit numbers that can be chosen which are even and greater than 3000, we need to consider the following:

Step 1: Determine the possibilities for the thousands place digit (the leftmost digit).

Since we want the number to be greater than 3000 and even, the thousands place digit must be 4, 5, or 6. Therefore, we have 3 possibilities for the thousands place digit.

Step 2: Determine the possibilities for the hundreds, tens, and units place digits (the remaining three digits).

For each of the thousands place digit possibilities, we have 5 remaining digits to choose from (since the digits are selected without replacement). Therefore, we have 5 possibilities for the hundreds place digit, 4 possibilities for the tens place digit, and 3 possibilities for the units place digit.

Step 3: Calculate the total number of possibilities.

To find the total number of possibilities, we multiply the number of possibilities for each digit place together.

Total number of even 4-digit numbers greater than 3000 = 3 (possibilities for the thousands place digit) * 5 (possibilities for the hundreds place digit) * 4 (possibilities for the tens place digit) * 3 (possibilities for the units place digit)

Therefore, the total number of 4-digit numbers that can be chosen, which are even and greater than 3000, is 3*5*4*3 = 180.

To answer these questions, we need to understand the concept of permutations and combinations.

1. How many 4-digit numbers can be selected that are greater than 3000?
We have 6 options for the thousands place (3, 4, 5, or 6), 5 options for the hundreds place (since we used one digit for the thousands place, 6 - 1 = 5 options remaining), 4 options for the tens place, and 3 options for the ones place. Therefore, the total number of 4-digit numbers greater than 3000 is calculated as follows:
Number of options = 6 * 5 * 4 * 3 = 360

2. How many even 4-digit numbers can be chosen?
To make a number even, the ones place must be one of the even digits (2, 4, 6). We can have 3 options for the ones place since it must be an even digit. Similar to the first question, we have 6 options for the thousands place, 5 options for the hundreds place, and 4 options for the tens place. Therefore, the total number of even 4-digit numbers is calculated as follows:
Number of options = 6 * 5 * 4 * 3 = 360

3. How many 4-digit numbers can be chosen that are even and greater than 3000?
To find the answer, we need to consider the intersection of the conditions for being even and being greater than 3000.
Since the ones place must be even, we have 3 options for it. The thousands place must be one of the digits 3, 4, 5, or 6, so we have 4 options for it. For the hundreds place, we can choose from the remaining 5 digits (since we already used one digit for the thousands place). Finally, we have 4 options for the tens place (since we used one digit for both the thousands and hundreds places). Therefore, the total number of 4-digit numbers that are even and greater than 3000 is calculated as follows:
Number of options = 4 * 5 * 4 * 3 = 240

In summary:
1. The number of 4-digit numbers greater than 3000 = 360
2. The number of even 4-digit numbers = 360
3. The number of 4-digit numbers that are even and greater than 3000 = 240