In an examination, 10,500 candidates appeared. The statistics show: passed in all five subjects = 5,685; passed in three subjects only = 1,498; passed in two subjects only = 1,250; passed in one subject only = 835; failed in English only = 78; failed in Maths only = 275; failed in Physics only = 149; failed in Chemistry only = 147; and failed in Biology only = 221. Based on this data, how many candidates passed in at least four subjects?

Introduction / Context:
This arithmetic reasoning problem uses examination statistics to test understanding of how passing and failing in individual subjects translate into passing in a certain number of subjects overall. The question asks for the number of candidates who passed in at least four subjects, which includes those who passed in all five as well as those who passed in exactly four subjects.

Given Data / Assumptions:

• Total candidates who appeared: 10,500.
• Passed all five subjects: 5,685.
• Passed exactly three subjects: 1,498.
• Passed exactly two subjects: 1,250.
• Passed exactly one subject: 835.
• Failed in English only: 78.
• Failed in Maths only: 275.
• Failed in Physics only: 149.
• Failed in Chemistry only: 147.
• Failed in Biology only: 221.

Concept / Approach:
A candidate who fails in exactly one subject must have passed the remaining four subjects. Therefore, the counts “failed in English only,” “failed in Maths only,” and so on all represent candidates who passed in exactly four subjects. The number of candidates who passed in at least four subjects equals the number who passed all five plus the number who passed exactly four. The problem therefore reduces to adding the five failure-only counts and then adding the count of those who passed all five subjects.

Step-by-Step Solution:
Step 1: Recognize that failing in English only means passing the other four subjects, and similarly for each other subject. Step 2: Compute the number of candidates who passed exactly four subjects by summing the failure-only counts: 78 + 275 + 149 + 147 + 221. Step 3: 78 + 275 = 353; 353 + 149 = 502; 502 + 147 = 649; 649 + 221 = 870. So 870 candidates passed exactly four subjects. Step 4: Candidates who passed in at least four subjects = passed in all five + passed in exactly four = 5,685 + 870. Step 5: 5,685 + 870 = 6,555.

Verification / Alternative check:
As a check, verify that no candidate is double counted in the failure-only categories. Each person recorded as “failed in X only” cannot be in any other single-failure list, because that would mean failure in more than one subject. Hence these five groups are disjoint and can be safely added. The result 870 represents all four-subject passers. Adding 5,685 five-subject passers gives 6,555, which matches one of the options, confirming consistency.

Why Other Options Are Wrong:
Option B 5,685 counts only those who passed all five subjects and ignores the four-subject passers. Option C 1,705 could come from incorrect partial addition of the data. Option D 870 counts only those who failed in exactly one subject, missing all candidates who passed in all five subjects.

Common Pitfalls:
Some learners mistakenly interpret “failed in English only” as failing in all subjects except English, which reverses the meaning. Another mistake is to assume that the numbers provided for passing in one, two, or three subjects are needed to compute the four-subject passers, when in fact the four-subject group is directly given via the single-failure counts. Keeping track of the exact language “failed in X only” versus “passed in X only” is crucial.

Final Answer:
The number of candidates who passed in at least four subjects is 6,555.