To find the probability of forming a 7-digit number from the digits 3, 3, 4, 4, 4, 5, 5 such that the number is divisible by 2, we will follow these steps:
### Step 1: Determine Total Outcomes
We first need to calculate the total number of 7-digit numbers that can be formed using the digits provided. The total number of arrangements of the digits can be calculated using the formula for permutations of multiset:
\[
\text{Total Outcomes} = \frac{7!}{2! \times 3! \times 2!}
\]
Where:
- \(7!\) is the factorial of the total number of digits,
- \(2!\) accounts for the two 3's,
- \(3!\) accounts for the three 4's,
- \(2!\) accounts for the two 5's.
Calculating this gives:
\[
7! = 5040
\]
\[
2! = 2, \quad 3! = 6
\]
\[
\text{Total Outcomes} = \frac{5040}{2 \times 6 \times 2} = \frac{5040}{24} = 210
\]
### Step 2: Determine Favorable Outcomes
Next, we need to find the number of favorable outcomes where the 7-digit number is divisible by 2. A number is divisible by 2 if its last digit is even. The only even digit available is 4.
Since we have three 4's, we can fix one 4 as the last digit. This leaves us with the digits: 3, 3, 4, 5, 5 (the remaining 6 digits).
Now, we calculate the arrangements of these 6 digits:
\[
\text{Favorable Outcomes} = \frac{6!}{2! \times 2!}
\]
Where:
- \(6!\) is the factorial of the remaining digits,
- \(2!\) accounts for the two 3's,
- \(2!\) accounts for the two 5's.
Calculating this gives:
\[
6! = 720
\]
\[
\text{Favorable Outcomes} = \frac{720}{2 \times 2} = \frac{720}{4} = 180
\]
### Step 3: Calculate Probability
Now, we can calculate the probability of forming a 7-digit number that is divisible by 2:
\[
\text{Probability} = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} = \frac{180}{210} = \frac{6}{7}
\]
### Final Answer
Thus, the probability that a 7-digit number formed from the digits 3, 3, 4, 4, 4, 5, 5 is divisible by 2 is:
\[
\frac{6}{7}
\]
