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Out of 1000 student 658 failed in the aggregate, 166 in the aggregate and in group-I 434 in aggregate and in group-II, 372 in group-I, 590 in group-II and 126 in both the groups. Find out how many failed in all the three.
- a)106
- b)224
- c)206
- d)464
Correct answer is option 'A'. Can you explain this answer?
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Out of 1000 student 658 failed in the aggregate, 166 in the aggregate ...
Given data:
- Total number of students = 1000
- Number of students failed in aggregate = 658
- Number of students failed in group-I = 434
- Number of students failed in group-II = 590
- Number of students failed in both group-I and group-II = 126
- Number of students failed in aggregate and group-I = 166
To find: Number of students who failed in all three groups
Solution:
We can use the formula of the inclusion-exclusion principle to solve this problem.
Total number of students failed in at least one group = Number of students failed in aggregate + Number of students failed in group-I + Number of students failed in group-II - Number of students failed in both group-I and group-II - Number of students failed in aggregate and group-I + Number of students failed in all three groups
Substituting the given values in the above formula, we get:
658 + 434 + 590 - 126 - 166 + Number of students failed in all three groups = Total number of students failed in at least one group
=> 1386 - 126 - 166 + Number of students failed in all three groups = Total number of students failed in at least one group
=> 1094 + Number of students failed in all three groups = Total number of students failed in at least one group
Total number of students failed in at least one group = 1000 - Number of students passed in all three groups
=> Total number of students failed in at least one group = 1000 - (Number of students who passed in aggregate - Number of students who failed in aggregate and group-I - Number of students who failed in all three groups)
Substituting the given values in the above formula, we get:
1094 + Number of students failed in all three groups = 1000 - (342 - 126 - Number of students failed in all three groups)
=> 1094 + Number of students failed in all three groups = 784 + Number of students failed in all three groups
=> Number of students failed in all three groups = 106
Therefore, the number of students who failed in all three groups is 106.
- Total number of students = 1000
- Number of students failed in aggregate = 658
- Number of students failed in group-I = 434
- Number of students failed in group-II = 590
- Number of students failed in both group-I and group-II = 126
- Number of students failed in aggregate and group-I = 166
To find: Number of students who failed in all three groups
Solution:
We can use the formula of the inclusion-exclusion principle to solve this problem.
Total number of students failed in at least one group = Number of students failed in aggregate + Number of students failed in group-I + Number of students failed in group-II - Number of students failed in both group-I and group-II - Number of students failed in aggregate and group-I + Number of students failed in all three groups
Substituting the given values in the above formula, we get:
658 + 434 + 590 - 126 - 166 + Number of students failed in all three groups = Total number of students failed in at least one group
=> 1386 - 126 - 166 + Number of students failed in all three groups = Total number of students failed in at least one group
=> 1094 + Number of students failed in all three groups = Total number of students failed in at least one group
Total number of students failed in at least one group = 1000 - Number of students passed in all three groups
=> Total number of students failed in at least one group = 1000 - (Number of students who passed in aggregate - Number of students who failed in aggregate and group-I - Number of students who failed in all three groups)
Substituting the given values in the above formula, we get:
1094 + Number of students failed in all three groups = 1000 - (342 - 126 - Number of students failed in all three groups)
=> 1094 + Number of students failed in all three groups = 784 + Number of students failed in all three groups
=> Number of students failed in all three groups = 106
Therefore, the number of students who failed in all three groups is 106.
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