Let's tackle this step-by-step.
### (1) Probability that the code will consist only of even digits
1. Identify the even digits and the total digits available:
- Total digits: {1, 2, 3, 4, 5, 6, 7, 8, 9} (9 digits)
- Even digits: {2, 4, 6, 8} (4 digits)
2. Determine the total number of four-digit codes:
- Since any digit can repeat, each of the four positions in the code can be any of the 9 digits.
- Total possible combinations = [tex]\( 9^4 = 6561 \)[/tex]
3. Determine the number of four-digit codes using only even digits:
- Each of the four positions must be one of the 4 even digits.
- Total even combinations = [tex]\( 4^4 = 256 \)[/tex]
4. Calculate the probability:
[tex]\[
\text{Probability of all even digits} = \frac{\text{Total even combinations}}{\text{Total possible combinations}} = \frac{256}{6561} \approx 0.03901844231062338
\][/tex]
Therefore, the probability that the code will consist only of even digits is approximately 0.0390 or 3.90%.
### (2) Probability that the digits of the code will all be different
1. Determine the total number of four-digit codes:
- As before, total possible combinations = [tex]\( 9^4 = 6561 \)[/tex]
2. Determine the number of four-digit codes with all different digits:
- We need to select 4 different digits out of 9 and then arrange them.
- The number of ways to choose 4 digits out of 9 without repetition and order them is given by permutations of 9 choose 4: [tex]\( P(9, 4) = \frac{9!}{(9-4)!} = 9 \times 8 \times 7 \times 6 = 3024 \)[/tex]
3. Calculate the probability:
[tex]\[
\text{Probability of all unique digits} = \frac{P(9, 4)}{\text{Total possible combinations}} = \frac{3024}{6561} \approx 0.4609053497942387
\][/tex]
Therefore, the probability that the digits of the code will all be different is approximately 0.4609 or 46.09%.
### (3) Probability that the code will consist of different even digits
1. Identify the even digits and the total digits available:
- Even digits: {2, 4, 6, 8} (4 digits)
2. Determine the total number of four-digit codes:
- Total possible combinations = [tex]\( 9^4 = 6561 \)[/tex]
3. Determine the number of four-digit combinations with different even digits:
- We need to choose 4 different even digits out of the 4 available and arrange them.
- The number of ways to do this is [tex]\( P(4, 4) = \frac{4!}{(4-4)!} = 4! = 24 \)[/tex]
4. Calculate the probability:
[tex]\[
\text{Probability of different even digits} = \frac{P(4, 4)}{\text{Total possible combinations}} = \frac{24}{6561} \approx 0.003657978966620942
\][/tex]
Therefore, the probability that the code will consist of different even digits is approximately 0.0037 or 0.37%.
In summary:
1. Probability of all even digits: 0.0390 or 3.90%
2. Probability of all unique digits: 0.4609 or 46.09%
3. Probability of different even digits: 0.0037 or 0.37%
