Number of different four digit numbers that may be formed using each of the digits 1 2 3 4 5 6 7 and 8 only once so that the number contains 4 is

Options

A. `"^8 P_4`
B. `"^8C_4`
C. `"^7C_3 . 4!`
D. none of these

Answer

To solve the problem of finding the number of different four-digit numbers that can be formed using the digits 1, 2, 3, 4, 5, 6, 7, and 8, with the condition that the digit '4' must be included, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Total Digits Available**:
We have the digits: 1, 2, 3, 4, 5, 6, 7, 8. This gives us a total of 8 digits.

2. **Select the Required Digits**:
Since we need to form a four-digit number that must include the digit '4', we will fix '4' in one of the four positions.

3. **Choose the Remaining Digits**:
After fixing '4', we need to choose 3 more digits from the remaining digits (1, 2, 3, 5, 6, 7, 8). This gives us 7 remaining digits to choose from.

4. **Calculate the Combinations**:
The number of ways to choose 3 digits from the remaining 7 digits can be calculated using the combination formula \( \binom{n}{r} \):
\[ \text{Number of ways to choose 3 digits} = \binom{7}{3} \]

5. **Calculate the Factorial for Arrangements**:
Once we have chosen the 3 additional digits along with '4', we will have a total of 4 digits. The number of ways to arrange these 4 digits is given by \( 4! \) (4 factorial).

6. **Combine the Results**:
The total number of different four-digit numbers is then calculated by multiplying the number of ways to choose the digits by the number of arrangements:
\[ \text{Total Numbers} = \binom{7}{3} \times 4! \]

7. **Calculate the Values**:
- Calculate \( \binom{7}{3} \):
\[ \binom{7}{3} = \frac{7!}{3!(7-3)!} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35 \]
- Calculate \( 4! \):
\[ 4! = 4 \times 3 \times 2 \times 1 = 24 \]
- Now multiply these results:
\[ \text{Total Numbers} = 35 \times 24 = 840 \]

### Final Answer:
The total number of different four-digit numbers that can be formed using the digits 1, 2, 3, 4, 5, 6, 7, and 8, such that the number contains '4', is **840**.