how many positive 4-digit numbers are there with an even digit in the hundreds position and an odd digit in the tens position?
a. 10,000
b. 5,040
c. 2,500
d. 2,250
To solve this problem, we can break it down into steps:
Step 1: Determine the number of possibilities for the even digit in the hundreds position.
Since the question specifies that the even digit goes in the hundreds position, we can choose from digits 0, 2, 4, 6, and 8. So there are 5 possibilities.
Step 2: Determine the number of possibilities for the odd digit in the tens position.
Since the question specifies that the odd digit goes in the tens position, we can choose from digits 1, 3, 5, 7, and 9. So there are 5 possibilities.
Step 3: Determine the number of possibilities for the remaining two digits in the units and thousands positions.
For the units position, we can choose any digit from 0 to 9, so there are 10 possibilities.
For the thousands position, we can choose any digit from 1 to 9 (since it cannot be 0), so there are 9 possibilities.
Step 4: Multiply the number of possibilities from each step together.
To get the total number of positive 4-digit numbers that satisfy the given conditions, we multiply the number of possibilities from each step:
5 (possibilities for the even digit in the hundreds position) ×
5 (possibilities for the odd digit in the tens position) ×
10 (possibilities for the units position) ×
9 (possibilities for the thousands position) =
5 × 5 × 10 × 9 = 2250
Therefore, the answer is d. 2,250.
To find the number of positive 4-digit numbers with an even digit in the hundreds position and an odd digit in the tens position, we can break down the problem into steps.
Step 1: Determine the number of choices for the hundreds digit
Since we need to have an even digit in the hundreds position, there are 5 choices: 0, 2, 4, 6, or 8.
Step 2: Determine the number of choices for the tens digit
Since we need to have an odd digit in the tens position, there are 5 choices: 1, 3, 5, 7, or 9.
Step 3: Determine the number of choices for the units and thousands digit
For the units digit, we have 10 choices (0-9), and for the thousands digit, we have 10 choices as well.
Step 4: Multiply the choices from each step
To get the total number of positive 4-digit numbers, we multiply the choices from each step together: 5 (choices for hundreds digit) × 5 (choices for tens digit) × 10 (choices for units digit) × 10 (choices for thousands digit) = 2,500.
Therefore, the correct answer is option c. 2,500.
1st digit has 9 choices
2nd has 5
3rd has 5
4th has 10
so, ...