A five digit number if formed by the digits 1,2,3,4,5,6 and 8. The probability that the number has even digit at both ends is

Options

A. `(3)/(7)`
B. `(4)/(7)`
C. `(2)/(7)`
D. none of these

Explanation

To solve the problem of finding the probability that a five-digit number formed from the digits 1, 2, 3, 4, 5, 6, and 8 has even digits at both ends, we can follow these steps:

### Step 1: Determine the Total Number of Five-Digit Numbers
We have 7 digits: 1, 2, 3, 4, 5, 6, and 8. We need to form a five-digit number using these digits. The total number of ways to select and arrange 5 digits from 7 is calculated as follows:

- For the first digit, we have 7 choices.
- For the second digit, we have 6 choices (one digit has already been used).
- For the third digit, we have 5 choices.
- For the fourth digit, we have 4 choices.
- For the fifth digit, we have 3 choices.

Thus, the total number of five-digit numbers is:
\[ 7 \times 6 \times 5 \times 4 \times 3 \]

### Step 2: Identify the Even Digits
The even digits available from our set are 2, 4, 6, and 8. There are 4 even digits in total.

### Step 3: Choose Even Digits for Both Ends
To satisfy the condition of having even digits at both ends, we can choose 2 even digits from the 4 available. The number of ways to choose 2 even digits can be calculated using combinations:
\[ \text{Number of ways to choose 2 even digits} = \binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4 \times 3}{2 \times 1} = 6 \]

### Step 4: Arrange the Chosen Even Digits
Each pair of chosen even digits can be arranged in 2 ways (one can be at the start and the other at the end). Therefore, the total arrangements for the ends is:
\[ 6 \times 2 = 12 \]

### Step 5: Fill the Middle Three Digits
After placing the even digits at both ends, we have 5 remaining digits (since we used 2). We can choose any 3 from these 5 remaining digits to fill the middle positions. The number of ways to arrange these 3 digits is:
\[ 5 \times 4 \times 3 \]

### Step 6: Calculate the Total Favorable Outcomes
Now, the total number of favorable outcomes (five-digit numbers with even digits at both ends) is:
\[ 12 \times (5 \times 4 \times 3) = 12 \times 60 = 720 \]

### Step 7: Calculate the Total Possible Outcomes
The total number of five-digit numbers we calculated in Step 1 is:
\[ 7 \times 6 \times 5 \times 4 \times 3 = 2520 \]

### Step 8: Calculate the Probability
The probability \( P \) that a randomly formed five-digit number has even digits at both ends is given by:
\[ P = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{720}{2520} \]

### Step 9: Simplify the Probability
We can simplify this fraction:
\[ P = \frac{720 \div 720}{2520 \div 720} = \frac{1}{3.5} = \frac{2}{7} \]

Thus, the probability that the number has even digits at both ends is:
\[ \frac{2}{7} \]