Consider seven digit number `x_1,x_2,...,x_7,` where `x_1,x_2,..,x_7 != 0` having the property that `x_4` is the greatest digit and digits towards the left and right of `x_4` are in decreasing order. Then total number of such numbers in which all digits are distinct is

Options

A. `""^(9)C_(7)*""^(6)C_(3)`
B. `""^(9)C_(5)+2""^(9)C_(4)`
C. `3""^(9)C_(4)`
D. `378`

Answer

To solve the problem of finding the total number of seven-digit numbers \( x_1, x_2, \ldots, x_7 \) such that all digits are distinct, none of the digits are zero, \( x_4 \) is the greatest digit, and the digits to the left and right of \( x_4 \) are in decreasing order, we can follow these steps:

### Step 1: Identify the possible values for \( x_4 \)
Since \( x_4 \) is the greatest digit and must be one of the digits from 1 to 9 (as none of the digits can be zero), we can choose \( x_4 \) from the set \( \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \).

### Step 2: Choose the digits for \( x_4 \)
Let’s denote the value of \( x_4 \) as \( d \). Since \( d \) is the greatest digit, it can take any value from 1 to 9. We will have 9 choices for \( d \).

### Step 3: Select the remaining digits
After choosing \( d \) for \( x_4 \), we need to select 6 more distinct digits from the remaining digits (which are from 1 to 9 excluding \( d \)). This gives us \( 8 \) remaining digits to choose from.

### Step 4: Choose 6 digits from the remaining 8
We need to choose 3 digits for the left side of \( x_4 \) (which will be \( x_1, x_2, x_3 \)) and 3 digits for the right side of \( x_4 \) (which will be \( x_5, x_6, x_7 \)). The number of ways to select 6 digits from the 8 remaining digits is given by \( \binom{8}{6} \) or equivalently \( \binom{8}{2} \).

### Step 5: Arrange the selected digits
Once we have selected the 6 digits, we need to arrange them in decreasing order. Since the digits on the left of \( x_4 \) must be in decreasing order, we can simply arrange the 3 digits chosen for the left side in 1 way (as they must be in decreasing order). The same applies to the 3 digits chosen for the right side.

### Step 6: Calculate the total combinations
Putting it all together, the total number of distinct seven-digit numbers can be calculated as follows:

\[ \text{Total Numbers} = \text{Choices for } x_4 \times \text{Ways to choose 6 digits} \]
\[ = 9 \times \binom{8}{6} = 9 \times \binom{8}{2} \]
\[ = 9 \times \frac{8 \times 7}{2 \times 1} = 9 \times 28 = 252 \]

### Final Answer
Thus, the total number of such seven-digit numbers is **252**.

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