A seven digit number is formed using digits 3,3,4,4,4,5,5. The probability, that number so formed is divisble by 2, is:

Options

A. `6/7`
B. `1/7`
C. `3/7`
D. `4/7`

Answer

To find the probability that a seven-digit number formed using the digits 3, 3, 4, 4, 4, 5, 5 is divisible by 2, we can follow these steps:

### Step 1: Determine the total number of seven-digit numbers that can be formed.
We have the digits: 3, 3, 4, 4, 4, 5, 5. The total number of arrangements of these digits can be calculated using the formula for permutations of multiset:

\[ \text{Total arrangements} = \frac{7!}{2! \cdot 3! \cdot 2!} \]

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:

\[ \text{Total arrangements} = \frac{5040}{2 \cdot 6 \cdot 2} = \frac{5040}{24} = 210 \]

### Step 2: Determine the favorable outcomes where the number is divisible by 2.
A number is divisible by 2 if its last digit is even. The only even digit we have is 4. Therefore, we can fix one of the 4's at the end of the number.

Now, we have the remaining digits: 3, 3, 4, 5, 5 (total of 6 digits). The number of arrangements of these 6 digits is:

\[ \text{Favorable arrangements} = \frac{6!}{2! \cdot 2!} \]

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:

\[ \text{Favorable arrangements} = \frac{720}{2 \cdot 2} = \frac{720}{4} = 180 \]

### Step 3: Calculate the probability.
The probability \(P\) that a randomly formed seven-digit number is divisible by 2 is given by the ratio of the number of favorable outcomes to the total outcomes:

\[ P = \frac{\text{Favorable arrangements}}{\text{Total arrangements}} = \frac{180}{210} \]

This simplifies to:

\[ P = \frac{6}{7} \]

### Final Answer
The probability that the number formed is divisible by 2 is:

\[ \frac{6}{7} \]