from the digits 0,1,2,3,4,5,6,7,8,9 find the number of
a) 4-digit numbers with repetition of digits
b) 4-digit numbers without repetition of digits
c) the number of odd 4-digit numbers with no repetition of digits
d) the number of even 4-digit numbers with no repetition of digits
We will be glad to criti9ue your thinking.
For (a), how many numbers are between 0 and 9999? Some have repeated digits.
These are not hard when you think about them
To find the number of 4-digit numbers with repetition or without repetition of digits, we can use the concept of permutations. Permutations deal with arranging objects in a specific order.
a) To find the number of 4-digit numbers with repetition of digits, we have 10 options for each digit (0-9) since repetition is allowed. We can use the formula for permutations with repetition, which is n^r, where n is the number of options available for each digit, and r is the number of digits. Therefore, the number of 4-digit numbers with repetition of digits is 10^4 = 10,000.
b) To find the number of 4-digit numbers without repetition of digits, we have 10 options for the first digit, 9 options for the second digit (as one option is already used), 8 options for the third digit, and 7 options for the fourth digit. We can use the formula for permutations without repetition, which is n! / (n - r)!. Here, n is the number of options available for the first digit, and r is the number of digits. Therefore, the number of 4-digit numbers without repetition of digits is 10! / (10 - 4)! = 10! / 6! = 10 x 9 x 8 x 7 = 5,040.
c) To find the number of odd 4-digit numbers with no repetition of digits, we need to ensure that the last digit is odd. As we have 5 odd digits from 0-9 (1, 3, 5, 7, 9), we can choose the last digit in 5 ways. The first digit can be any of the remaining 9 digits (excluding 0 and the digit already selected for the last digit), the second digit can be any of the remaining 8 digits, and the third digit can be any of the remaining 7 digits. Therefore, the total number of odd 4-digit numbers with no repetition of digits is 5 x 9 x 8 x 7 = 2,520.
d) To find the number of even 4-digit numbers with no repetition of digits, we need to ensure that the last digit is even (excluding 0). As we have 4 even digits from 0-9 (0, 2, 4, 6), we can choose the last digit in 4 ways. The first digit can be any of the remaining 9 digits (excluding the digit already selected for the last digit), the second digit can be any of the remaining 8 digits, and the third digit can be any of the remaining 7 digits. Therefore, the total number of even 4-digit numbers with no repetition of digits is 4 x 9 x 8 x 7 = 2,016.