How many 7 digit even numbers can be formed using all of the digits 0,1,1,2,3,4,5?
The answer is 1320, but how do I get there? Thanks for the help!!
To find the number of 7-digit even numbers that can be formed using the digits 0, 1, 1, 2, 3, 4, and 5, we can break down the process into several steps:
Step 1: Count the total number of 7-digit numbers that can be formed using all the given digits.
Step 2: Determine the number of those 7-digit numbers that are even.
Let's go step by step.
Step 1: Count the total number of 7-digit numbers that can be formed using all the given digits.
Since we are forming a 7-digit number, the first digit should be non-zero (as we cannot have leading zeros). Therefore, there are 6 options for the first digit (1, 1, 2, 3, 4, or 5).
For the second digit, any of the remaining 6 digits can be used (including 0), as repetitions are allowed.
Similarly, for the third digit, there are still 6 options.
For the fourth digit, we have 5 options left (excluding the number used in the first digit).
For the fifth digit, we also have 5 options left (excluding the two numbers used in the first and fourth digits).
For the sixth digit, there are 4 options left (excluding the three numbers used in the first, fourth, and fifth digits).
Finally, for the seventh digit, only 1 option is left (as we would have used all the other numbers).
To calculate the total number of possibilities, we multiply all these options together: 6 x 6 x 6 x 5 x 5 x 4 x 1 = 6^3 x 5^2 x 4 = 86400.
Step 2: Determine the number of 7-digit numbers that are even.
For a number to be even, the units digit (seventh digit) must be 0, 2, 4, or 5 since the remaining possible digits (1, 1, 2, 3, and 4) are all odd.
We already know that the seventh digit will be 0 as we have exhausted all the other numbers.
For the first digit, we have 5 options left (excluding the used number in the seventh digit).
For the second digit, there are still 6 options.
Similarly, for the third digit, there are 6 options.
For the fourth digit, there are 5 options.
For the fifth digit, there are 5 options.
For the sixth digit, there are 4 options.
To calculate the total number of even numbers, we multiply all these options together: 5 x 6 x 6 x 5 x 5 x 4 x 1 = 5^2 x 6^2 x 4 = 3600.
Therefore, the number of 7-digit even numbers that can be formed using all the digits 0, 1, 1, 2, 3, 4, and 5 is 3600.
I apologize for the confusion earlier. The previous answer of 1320 was incorrect. The correct answer is 3600.