Let's solve this step by step.

Carter's locker combination consists of three integers that:

1. Are prime numbers.
2. Sum to 50.
3. Are arranged in decreasing order.

Step 1: Identify prime numbers less than 50

Prime numbers less than 50 are: $2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47$

Step 2: Find a combination of three primes whose sum is 50

We need to find three prime numbers from the list such that their sum is 50, and they are in decreasing order.

Let's start by trying different combinations:

• The largest prime number less than 50 is 47. If we take 47, the remaining two numbers must sum to $50 - 47 = 3$, but no two primes sum to 3, so 47 is not part of the combination.
• Next, try 43. If we take 43, the remaining two numbers must sum to $50 - 43 = 7$. The prime numbers that sum to 7 are $5 + 2$.

So, the combination is $43, 5, 2$, which are in decreasing order.

Step 3: Verify the sum

$43 + 5 + 2 = 50$ The sum is correct.

Conclusion:

The last integer in Carter's locker combination is 2.

---

Would you like more details on this solution or have any questions?

Here are five related questions to expand your understanding:

1. What are the properties of prime numbers?
2. How can you verify if a number is prime?
3. What other sets of prime numbers can sum to a different total?
4. Can you create a similar problem with a sum of 60 using primes?
5. How would the solution change if one number in the combination had to be even?

Tip: Prime numbers are only divisible by 1 and themselves, which makes them fundamental in number theory and cryptography!