How many candidates passed in two or more subjects?

Given :

Total candidates = 500

Failed in English only = 30

Failed in Hindi only = 75

Failed in Mathematics only = 50

Failed in English and Hindi = 15

Failed in Hindi and Mathematics = 17

Failed in Mathematics and English = 17

Failed in all three = 5

Concept used:

Candidates who failed in at least one subject are counted by summing those who failed in only one subject, those who failed in exactly two subjects, and those who failed in all three. However, remember that those who failed in exactly two subjects and all three subjects get counted extra times in the "only" categories, so we adjust for this by adding back those numbers.

Calculations :

So, the calculation goes as follows:

Total failed = (30 + 75 + 50) + (15 + 17 + 17) - 2 x 5

The subtraction by 2 x 5 is essential because the 5 candidates who failed in all three were counted three times in the single failures and once more in the double failures, requiring us to subtract them twice to adjust for the overcounting.

Calculating the number:

Total failed = 155 + 49 - 10 = 194

Therefore, the number of candidates passing in  subjects is the total number of students minus the number who failed any subject:

Passed in all subjects = Total candidates - Total failed = 500 - 194 = 306

Passed in two or more subjects =  306 + candidates failed in only one subject = 306 + 30 + 75 + 50 = 461

So, the correct answer is option 1) 461.