Counting Four-Digit Numbers with Even Digits
The question asks us to find the total number of four-digit natural numbers where all the digits used are even.
Understanding the Requirements
A natural number is a positive whole number (1, 2, 3, ...). A four-digit number is a natural number between 1000 and 9999, inclusive.
The digits that are considered even are: 0, 2, 4, 6, 8. There are 5 even digits in total.
Applying Constraints to Digits
Let's consider the four positions of a four-digit number:
- Thousands place
- Hundreds place
- Tens place
- Units place
We need to determine the number of choices for each place, keeping in mind the constraints:
- The number must be a four-digit number, meaning the first digit (thousands place) cannot be 0.
- All digits must be even (from the set {0, 2, 4, 6, 8}).
Calculating Choices for Each Digit
- Thousands place: This digit must be even, but it cannot be 0. So, the possible choices are {2, 4, 6, 8}. This gives us 4 possible choices.
- Hundreds place: This digit can be any of the even digits. The possible choices are {0, 2, 4, 6, 8}. This gives us 5 possible choices.
- Tens place: Similar to the hundreds place, this digit can be any of the even digits {0, 2, 4, 6, 8}. This gives us 5 possible choices.
- Units place: This digit can also be any of the even digits {0, 2, 4, 6, 8}. This gives us 5 possible choices.
Total Count Calculation
To find the total number of such four-digit natural numbers, we multiply the number of choices for each digit together. This is based on the fundamental principle of counting.
Total count = (Choices for Thousands) \(\times\) (Choices for Hundreds) \(\times\) (Choices for Tens) \(\times\) (Choices for Units)
Total count = \(4 \times 5 \times 5 \times 5\)
Total count = \(4 \times (5^3)\)
Total count = \(4 \times 125\)
Total count = 500
Conclusion
Therefore, there are 500 four-digit natural numbers where all the digits are even.
Similar Questions
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- A man has \(7\) relatives (\(4\) women and \(3\) men). His wife also has \(7\) relatives (\(3\) women and \(4\) men). In how many ways can they invite \(3\) women and \(3\) men so that \(3\) of them are man's relatives and \(3\) of them are his wife's relatives?
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If the number of selections of r as well as (n + r) things from 5n different things are equal, then what is the value of r?
Important Questions from Permutation and Combination
The following figure is made with nine squares of length 1 cm each.
A student is standing at point P. The total number of shortest paths to reach point Q from point P, if the student walks along the boundaries of the squares only, is:
Consider the following statements (all the angles are measured in degrees) :
1. If two supplementary angles are $4x + 40$ and $10x$, then $x$ is;
2. The complementary angle of $\frac{4}{5} \times 45$ is :
3. If two complementary angles are in the ratio of $4 : 5$, then the smallest angle is :
Which of the following is correct?
Codes:
1 2 3- There are 40 persons in a palace. If every person shakes hands with every other person, what will be the total number of handshakes?
- How many words can be formed with or without meaning by using all letters of the word DELHI?
A rectangular wall is divided into four squares of equal size where there are two rows each having two squares. The top left square is coloured with green. If, including green, there are three colours available and each square is coloured using any one of these three colours such that no two adjacent squares get painted with the same colour; then how many colour combinations are possible ?
