How many numbers can be formed with the digits 2,4,4,3,7 which are greater than 40000?

Answer

To solve the problem of how many numbers can be formed with the digits 2, 4, 4, 3, and 7 that are greater than 40000, we can follow these steps:

### Step 1: Identify the possible leading digits
Since we need to form numbers greater than 40000, the leading digit (the first digit) must be either 4 or 7.

### Step 2: Case 1 - Leading digit is 4
If the leading digit is 4, the remaining digits to arrange are 2, 4, 3, and 7.

#### Step 2.1: Count arrangements of remaining digits
We have 4 digits (2, 4, 3, 7) to arrange. The number of arrangements of these 4 digits is calculated using the factorial of the number of digits:
\[ \text{Arrangements} = 4! = 24 \]

### Step 3: Case 2 - Leading digit is 7
If the leading digit is 7, the remaining digits to arrange are 2, 4, 4, and 3.

#### Step 3.1: Count arrangements of remaining digits
Here, we have 4 digits (2, 4, 4, 3) to arrange. Since the digit 4 is repeated, we need to divide by the factorial of the count of the repeated digit:
\[ \text{Arrangements} = \frac{4!}{2!} = \frac{24}{2} = 12 \]

### Step 4: Total arrangements
Now, we sum the arrangements from both cases:
\[ \text{Total arrangements} = \text{Arrangements with leading 4} + \text{Arrangements with leading 7} \]
\[ \text{Total arrangements} = 24 + 12 = 36 \]

### Final Answer
Thus, the total number of numbers that can be formed with the digits 2, 4, 4, 3, and 7 that are greater than 40000 is **36**.

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