In a semester examination there are 5 different subjects. A particular student…
In a semester examination there are 5 different subjects. A particular student is required to pass all the 5 subjects. Then the number of ways he can fail is
Answer: D. 31 — Concept: For a set of n distinct items, the total number of ways to choose zero or more of them (i.e., the number of subsets, including the empty one) is 2n.…
- A.
5!
- B.
5
- C.
1
- D.
31
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Correct answer: D
Concept: For a set of n distinct items, the total number of ways to choose zero or more of them (i.e., the number of subsets, including the empty one) is 2n. Since exactly one of these subsets is empty, the number of ways to choose a NON-EMPTY subset — equivalently, the sum of combinations nC1 + nC2 + … + nCn — equals 2n − 1.
Here, 'failing in one or more subjects' out of the 5 subjects means choosing a non-empty subset of the 5 subjects to fail in — so n = 5.
Group the outcomes by how many subjects are failed: exactly 1 subject → 5C1 = 5 ways; exactly 2 → 5C2 = 10 ways; exactly 3 → 5C3 = 10 ways; exactly 4 → 5C4 = 5 ways; all 5 → 5C5 = 1 way.
Add these mutually exclusive cases: 5 + 10 + 10 + 5 + 1 = 31.
Cross-check: Each of the 5 subjects is independently passed or failed, giving 25 = 32 total pass/fail outcomes in all. Exactly one of these — the outcome where all 5 are passed — is NOT a case of 'failing in at least one subject', so the count is 32 − 1 = 31, matching the sum above.