The digits 1, 2, 3, 4, 5, 6, 7, 8, and 9 are used to generate a four-digit code randomly. Any digit may be used any number of times.

1. Calculate the probability that the code will consist only of even digits.
2. Calculate the probability that the digits of the code will all be different.
3. Calculate the probability that the code will consist of different even digits.



Answer :

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%