Let A be the set of 4-digit numbers `a_1 a_2 a_3 a_4` where `a_1 > a_2 > a_3 > a_4`, then `n(A)` is equal to

Options

A. 126
B. `.^(10)C_(4)`
C. 210
D. None of these

Answer

To solve the problem, we need to determine the number of 4-digit numbers \( a_1 a_2 a_3 a_4 \) such that \( a_1 > a_2 > a_3 > a_4 \).

### Step-by-Step Solution:

1. **Understanding the Constraints**:
- We want to form a 4-digit number where each digit is distinct and arranged in strictly decreasing order. This means that we cannot have any repeated digits and each digit must be less than the one before it.

2. **Choosing Digits**:
- The digits available for our 4-digit number are from the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. This gives us a total of 10 digits to choose from.

3. **Selecting 4 Distinct Digits**:
- Since \( a_1 \) must be the largest digit and \( a_4 \) must be the smallest, we need to choose 4 distinct digits from the 10 available. The order in which we select the digits does not matter because they will be arranged in decreasing order.

4. **Calculating Combinations**:
- The number of ways to choose 4 digits from 10 is given by the combination formula \( \binom{n}{r} \), where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose. Here, \( n = 10 \) and \( r = 4 \).
\[ \binom{10}{4} = \frac{10!}{4!(10-4)!} = \frac{10!}{4!6!} \]

5. **Simplifying the Calculation**:
- We can simplify \( \binom{10}{4} \):
\[ \binom{10}{4} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} \]
- Calculating the numerator:
\[ 10 \times 9 = 90 \]
\[ 90 \times 8 = 720 \]
\[ 720 \times 7 = 5040 \]
- Calculating the denominator:
\[ 4 \times 3 = 12 \]
\[ 12 \times 2 = 24 \]
\[ 24 \times 1 = 24 \]
- Now, dividing the results:
\[ \frac{5040}{24} = 210 \]

6. **Conclusion**:
- Therefore, the number of 4-digit numbers \( n(A) \) such that \( a_1 > a_2 > a_3 > a_4 \) is \( 210 \).

### Final Answer:
\[ n(A) = 210 \]