If m denotes the number of 5 digit numbers if each successive digits are in their descending order of magnitude and n is the corresponding figure, when the digits are in their ascending order of magnitude then (m-n) has the value (

Options

A. `2.^(10)C_(5)`
B. `.^(10)C_(4)`
C. `.^(9)C_(2)`
D. `.^(9)C_(5)`

Answer

To solve the problem, we need to determine the values of \( m \) and \( n \) and then find \( m - n \).

### Step 1: Determine \( m \)
- \( m \) represents the number of 5-digit numbers where each successive digit is in descending order.
- We have the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 (a total of 10 digits).
- Since we are forming a 5-digit number, we can choose any 5 digits from these 10 digits. However, we cannot have 0 as the leading digit, so we will only consider combinations that do not start with 0.
- The number of ways to choose 5 digits from 10 is given by the combination formula \( \binom{10}{5} \).

Thus,
\[ m = \binom{10}{5} \]

### Step 2: Determine \( n \)
- \( n \) represents the number of 5-digit numbers where each successive digit is in ascending order.
- In this case, we can use the digits 1 to 9 (since 0 cannot be the leading digit).
- We can choose any 5 digits from the 9 digits (1 to 9).
- The number of ways to choose 5 digits from these 9 digits is given by \( \binom{9}{5} \).

Thus,
\[ n = \binom{9}{5} \]

### Step 3: Calculate \( m - n \)
Now we need to find \( m - n \):
\[ m - n = \binom{10}{5} - \binom{9}{5} \]

### Step 4: Simplify the expression
Using the property of combinations, we know:
\[ \binom{n}{r} = \binom{n-1}{r} + \binom{n-1}{r-1} \]
Applying this property here:
\[ \binom{10}{5} = \binom{9}{5} + \binom{9}{4} \]
Thus,
\[ m - n = \left( \binom{9}{5} + \binom{9}{4} \right) - \binom{9}{5} = \binom{9}{4} \]

### Final Answer
So, the value of \( m - n \) is:
\[ m - n = \binom{9}{4} \]