A three digit numbers is equal to the sum of the factorial of their digits . If the sum of all such three digit numbers is `lamda` then find the sum of digit of `lamda.`

Answer

To solve the problem, we need to find all three-digit numbers that are equal to the sum of the factorial of their digits. Let's denote a three-digit number as \(abc\), where \(a\), \(b\), and \(c\) are its digits. The equation we need to satisfy is:

\[ 100a + 10b + c = a! + b! + c! \]

### Step 1: Determine the range of digits
Since \(a\) is the hundreds digit, it must be between 1 and 9 (inclusive). The digits \(b\) and \(c\) can be between 0 and 9 (inclusive). However, we need to consider the factorial values:

- \(0! = 1\)
- \(1! = 1\)
- \(2! = 2\)
- \(3! = 6\)
- \(4! = 24\)
- \(5! = 120\)
- \(6! = 720\)
- \(7! = 5040\) (too large for a three-digit number)
- \(8! = 40320\) (too large for a three-digit number)
- \(9! = 362880\) (too large for a three-digit number)

Thus, the maximum digit we can use is 5, since \(6!\) and higher will exceed three digits.

### Step 2: Calculate factorials for digits 0 to 5
We will calculate the factorials for digits 0 to 5:

- \(0! = 1\)
- \(1! = 1\)
- \(2! = 2\)
- \(3! = 6\)
- \(4! = 24\)
- \(5! = 120\)

### Step 3: Check three-digit numbers
We can now check each three-digit number from 100 to 999 to see if it equals the sum of the factorials of its digits.

1. **For \(145\)**:
- Digits: \(1, 4, 5\)
- Calculation:
\[ 1! + 4! + 5! = 1 + 24 + 120 = 145 \]
- This satisfies the condition.

2. **For other numbers**:
- You can check other combinations of digits (e.g., \(1, 2, 3\) or \(2, 3, 4\)) but they will not satisfy the equation as shown in the video transcript.

After checking all combinations, we find that **145** is the only three-digit number that satisfies the condition.

### Step 4: Calculate the sum of all such numbers
Since \(145\) is the only number we found, we have:

\[ \lambda = 145 \]

### Step 5: Find the sum of the digits of \(\lambda\)
Now we need to find the sum of the digits of \(145\):

\[ 1 + 4 + 5 = 10 \]

### Final Answer
Thus, the sum of the digits of \(\lambda\) is:

\[ \boxed{10} \]