Evaluate the integer expression −261 + (−380) − (−521) + 821 − (−121) and find its exact numerical value.

Introduction / Context:
This is a straightforward arithmetic problem involving positive and negative integers. It tests your ability to handle addition and subtraction of signed numbers correctly, especially when multiple minus signs and brackets appear in the same expression.

Given Data / Assumptions:

• Expression: −261 + (−380) − (−521) + 821 − (−121).
• All numbers are integers.
• Standard rules of integer addition and subtraction apply.

Concept / Approach:
When dealing with signed numbers, we interpret addition of a negative number as subtraction and subtraction of a negative number as addition. We can simplify the series of operations by removing brackets and rewriting the expression in terms of plus and minus signs clearly, then combining the positive and negative contributions.

Step-by-Step Solution:
Step 1: Start with the given expression: −261 + (−380) − (−521) + 821 − (−121). Step 2: Remove brackets, keeping track of signs: = −261 − 380 + 521 + 821 + 121. Step 3: Group positive and negative terms separately. Positive terms: 521 + 821 + 121. Negative terms: −261 and −380. Step 4: Add positive terms: 521 + 821 = 1,342. Step 5: Then add 121: 1,342 + 121 = 1,463. Step 6: Add negative terms: −261 − 380 = −641. Step 7: Combine positive and negative totals: 1,463 + (−641) = 1,463 − 641. Step 8: Subtract: 1,463 − 641 = 822. Step 9: Therefore, the value of the expression is 822.

Verification / Alternative check:
You can also compute step by step from left to right: −261 − 380 = −641; then −641 − (−521) becomes −641 + 521 = −120; then −120 + 821 = 701; and finally 701 − (−121) becomes 701 + 121 = 822. This alternative calculation again yields 822, confirming the result.

Why Other Options Are Wrong:
Values like 833, 825, 800 and 812 result from arithmetic mistakes such as misadding the positives, forgetting to change a minus minus to plus, or mishandling one of the intermediate sums. A correct and careful calculation as shown leads uniquely to 822.

Common Pitfalls:
Typical errors include forgetting that subtracting a negative number is equivalent to adding a positive number, or incorrectly combining numbers with different signs. Keeping track of signs by rewriting the entire expression without brackets and then grouping positive and negative terms separately helps avoid mistakes.

Final Answer:
The exact value of the expression is 822.