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 have done, or shown you how to do, several of these probability questions for you; yet you persist in not showing any work, and reposting old questions. If you are not willing to make an attempt at learning the subject, this is not a place for you to seek help.

To find the answers to these questions, we can use the concepts of permutations and combinations.

a) 4-digit numbers with repetition of digits:
In this case, each digit can be selected from 0 to 9, and repetition is allowed.

There are 10 choices for each of the four digits of the 4-digit number. Therefore, the total number of 4-digit numbers with repetition would be 10^4 = 10,000.

b) 4-digit numbers without repetition of digits:
In this case, we cannot repeat any digit.

To calculate the number of 4-digit numbers without repetition, we need to count the number of ways we can select 4 different digits from the given set of digits (0 to 9). This can be calculated using combinations.

The formula for combinations is:
C(n, r) = n! / (r! * (n-r)!)

In this case, n = 10 (total number of digits) and r = 4 (number of digits to be selected). So, the number of 4-digit numbers without repetition would be C(10, 4):
C(10, 4) = 10! / (4! * (10-4)!)
= 10! / (4! * 6!)
= (10 * 9 * 8 * 7 * 6!) / (4! * 6!)
= (10 * 9 * 8 * 7) / (4 * 3 * 2 * 1)
= 210.

Therefore, there are 210 4-digit numbers without repetition.

c) The number of odd 4-digit numbers with no repetition of digits:
To find this, we need to consider two conditions:
1. The first digit must be odd (non-zero).
2. The last digit must be odd (1, 3, 5, 7, or 9).

The first digit can be selected in 5 ways (from 1, 3, 5, 7, or 9).
The last digit can be selected in 5 ways (from 1, 3, 5, 7, or 9).
The remaining two digits can be selected from the remaining 8 digits (0 and the remaining even digits).

Therefore, the total number of odd 4-digit numbers without repetition would be 5 * 8 * 7 * 5 = 1,400.

d) The number of even 4-digit numbers with no repetition of digits:
To find this, we need to consider two conditions:
1. The first digit must be even (0, 2, 4, 6, or 8).
2. The last digit must be even (0, 2, 4, 6, or 8).

The first digit can be selected in 5 ways (from 0, 2, 4, 6, or 8).
The last digit can be selected in 4 ways (from 0, 2, 4, 6, or 8).
The remaining two digits can be selected from the remaining 7 digits (1 and the remaining odd digits).

Therefore, the total number of even 4-digit numbers without repetition would be 5 * 7 * 6 * 4 = 840.

In summary:
a) 4-digit numbers with repetition of digits = 10,000
b) 4-digit numbers without repetition of digits = 210
c) The number of odd 4-digit numbers with no repetition of digits = 1,400
d) The number of even 4-digit numbers with no repetition of digits = 840