To solve this question, we need to find the smallest prime number that can be expressed as the sum of three different prime numbers. Let's go through the options given and check each one.

Step-by-Step Analysis:

1. Identify Prime Numbers: The smallest primes are 2, 3, 5, 7, 11, etc.

2. Testing Each Option:

   • Option (A) 10: $2 + 3 + 5 = 10$
     • 10 is not a prime number, so this option is invalid.
   • Option (B) 15: $2 + 5 + 7 = 14$
     • 14 is not a prime number, so this option is invalid.
   • Option (C) 17: $2 + 3 + 11 = 17$
     • 17 is a prime number, so this option is valid.
   • Option (D) 19: $3 + 5 + 11 = 19$
     • 19 is a prime number, so this option is also valid, but it's larger than 17.
   • Option (E) 23: $5 + 7 + 11 = 23$
     • 23 is a prime number, but it's also larger than 17.

3. Conclusion: Among the options, 17 is the smallest prime number that can be written as the sum of three different prime numbers (2, 3, and 11).

Answer:

The correct answer is (C) 17.

Would you like further details on any steps?

Here are five related questions for further practice:

1. What is the smallest odd prime that can be written as the sum of three different prime numbers?
2. Find the smallest even number that can be expressed as the sum of three prime numbers.
3. Can any prime number be written as the sum of two prime numbers? Why or why not?
4. What is the smallest prime number that can be written as the sum of four different prime numbers?
5. Are there any patterns in prime sums that help predict other combinations of prime sums?

Tip: When testing multiple options, break down each one step-by-step and eliminate based on conditions provided.