To solve the problem of forming two-digit numbers using the digits 0, 1, 2, 3, 4, and 5 under the given conditions, let's go through each event one by one:
### Definition:
Let's denote by [tex]\( N \)[/tex] the two-digit number of interest. Because [tex]\( N \)[/tex] is a two-digit number, its tens place cannot be zero.
Digits:
[tex]\[ \text{digits} = \{0, 1, 2, 3, 4, 5\} \][/tex]
### Condition for event A: The number formed is even.
For the number [tex]\( N \)[/tex] to be even, its unit's place must be one of the even digits from the given set.
The even digits available are:
[tex]\[ \{0, 2, 4\} \][/tex]
Possible choices for the tens place (cannot be 0) are:
[tex]\[ \{1, 2, 3, 4, 5\} \][/tex]
Number of choices for units place = 3 (0, 2, or 4).
Number of choices for tens place = 5 (1, 2, 3, 4, 5).
Thus, the number of possible two-digit even numbers is:
[tex]\[ 5 \text{ (choices for tens place)} \times 3 \text{ (choices for units place)} = 15 \][/tex]
There are [tex]\( 15 \)[/tex] two-digit even numbers.
### Condition for event B: The number formed is divisible by 3.
For [tex]\( N \)[/tex] to be divisible by 3, the sum of its digits must be divisible by 3.
We list all pairs [tex]\((a, b)\)[/tex] where [tex]\( a + b \)[/tex] mod 3 = 0:
- Sum that equals 3:
- (1,2), (2,1)
- Sum that equals 6:
- (1,5), (2,4), (3,3), (4,2), (5,1)
- Sum that equals 9:
- (3,6), (4,5), (5,4)
Considering digit sums and ensuring non-zero tens place, the pairs are:
- (1,2), (1,5), (2,1), (2,4), (3,3), (4,2), (4,5), (5,1), (5,4), (6,3)
Thus, the number of such pairs is 10.
Therefore, there are [tex]\( 10 \)[/tex] two-digit numbers divisible by 3.
### Condition for event C: The number formed is greater than 50.
For the number [tex]\( N \)[/tex] to be greater than 50, the tens digit must be 5:
[tex]\[ \{5\} \][/tex]
Unit place can be:
[tex]\[ \{0, 1, 2, 3, 4, 5\} \][/tex]
But for numbers only greater than 50:
- The valid pairs are: (5,1), (5,2), (5,3), (5,4), (5,5).
We have 5 possible pairs that make the number greater than 50:
(5,1), (5,2), (5,3), (5,4), (5,5)
Thus, there are [tex]\( 5 \)[/tex] two-digit numbers greater than 50.
### Summary:
- Number of two-digit even numbers: 15
- Number of two-digit numbers divisible by 3: 10
- Number of two-digit numbers greater than 50: 5
