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. 84
C. 210
D. None of these

Explanation

To solve the problem, we need to find the number of 4-digit numbers \( a_1 a_2 a_3 a_4 \) such that \( a_1 > a_2 > a_3 > a_4 \). Here’s the step-by-step solution:

### Step 1: Understand the constraints
We are tasked with forming a 4-digit number where each digit is distinct and arranged in strictly decreasing order. This means that we can only use digits from the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.

### Step 2: Choose 4 distinct digits
Since we need to form a 4-digit number, we will choose 4 distinct digits from the available 10 digits (0 to 9). The order of the digits will be determined by the condition \( a_1 > a_2 > a_3 > a_4 \).

### Step 3: Calculate the number of ways to choose 4 digits
The number of ways to choose 4 digits from a set of 10 can be calculated using the combination formula:
\[ nCr = \frac{n!}{r!(n-r)!} \]
In our case, \( n = 10 \) and \( r = 4 \):
\[ 10C4 = \frac{10!}{4!(10-4)!} = \frac{10!}{4! \cdot 6!} \]

### Step 4: Simplify the combination
Now we simplify \( 10C4 \):
\[ 10C4 = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} \]
Calculating the numerator:
\[ 10 \times 9 = 90 \]
\[ 90 \times 8 = 720 \]
\[ 720 \times 7 = 5040 \]
Now calculating the denominator:
\[ 4 \times 3 = 12 \]
\[ 12 \times 2 = 24 \]
\[ 24 \times 1 = 24 \]
Now divide the numerator by the denominator:
\[ 10C4 = \frac{5040}{24} = 210 \]

### Step 5: Conclusion
Thus, the number of 4-digit numbers \( a_1 a_2 a_3 a_4 \) such that \( a_1 > a_2 > a_3 > a_4 \) is \( n(A) = 210 \).

### Final Answer
\[ \boxed{210} \]