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Out of 1000 students 658 failed in aggregate, 166 in the aggregate and in group-1,434 in aggregate and in group-2,372 in group-1,590 in group-2 and 126 in both groups. Find out how many failed in all three?
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Out of 1000 students 658 failed in aggregate, 166 in the aggregate and...
Problem: Out of 1000 students, 658 failed in aggregate, 166 in the aggregate and in group-1, 434 in aggregate and in group-2, 372 in group-1, 590 in group-2, and 126 in both groups. Find out how many failed in all three?

Solution:
To find out the number of students who failed in all three, we need to use the principle of inclusion and exclusion.

Step 1: Find the total number of students who failed in group-1 and/or group-2.
- Number of students who failed in group-1 = 372 + 166 - 126 = 412
- Number of students who failed in group-2 = 590 + 434 - 126 = 898
- Total number of students who failed in group-1 and/or group-2 = 412 + 898 = 1310

Step 2: Find the total number of students who failed in aggregate but not in group-1 or group-2.
- Number of students who failed in aggregate but not in group-1 or group-2 = 658 - 1310 = 0 (since the result is negative, it means that all the students who failed in aggregate also failed in either group-1 or group-2, or both)

Step 3: Find the number of students who failed in all three.
- Number of students who failed in all three = Number of students who failed in group-1 and group-2 - Number of students who failed in aggregate but not in group-1 or group-2 - Number of students who failed in both group-1 and group-2
- Number of students who failed in all three = 126

Therefore, 126 students failed in all three groups.
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Out of 1000 students 658 failed in aggregate, 166 in the aggregate and in group-1,434 in aggregate and in group-2,372 in group-1,590 in group-2 and 126 in both groups. Find out how many failed in all three?
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