A student has to opt for 2 subjects out of 5 subjects for a course, namely, Commerce, Economics, Statistics, Mathematics I and Mathematics II. Mathematics II can be offered only if Mathematics I is also opted. The number of different combinations of two subjects which can be opted is

(a) 5
(b) 6
(c) 7 ✓ UPSC's answer
(d) 8

Why the answer is (c)

• The total number of ways to choose 2 subjects from 5 without restrictions is C(5,2) = 10.

• The restriction states that Mathematics II (M2) can only be chosen if Mathematics I (M1) is also chosen.

• Therefore, the combination {M2, Commerce}, {M2, Economics}, {M2, Statistics}, and {M2, Mathematics I} are the only valid pairs involving M2. Wait, {M2, M1} is valid. {M2, Commerce} is invalid because M1 is not present. {M2, Economics} is invalid. {M2, Statistics} is invalid.

• Let's list all valid pairs:

1. {Commerce, Economics}

2. {Commerce, Statistics}

3. {Commerce, Mathematics I}

4. {Commerce, Mathematics II} -> Invalid (M1 missing)

5. {Economics, Statistics}

6. {Economics, Mathematics I}

7. {Economics, Mathematics II} -> Invalid (M1 missing)

8. {Statistics, Mathematics I}

9. {Statistics, Mathematics II} -> Invalid (M1 missing)

10. {Mathematics I, Mathematics II} -> Valid (M1 is present)

• The valid combinations are: {C, E}, {C, S}, {C, M1}, {E, S}, {E, M1}, {S, M1}, {M1, M2}. This totals 7 combinations.

• Thus, the correct number of different combinations is 7.

Why the other options are wrong

(a) 5: Option (a) is incorrect because it undercounts the valid combinations by excluding valid pairs that do not involve Mathematics II or by misapplying the restriction.
(b) 6: Option (b) is incorrect because it misses one valid combination, likely the pair {Mathematics I, Mathematics II} or one of the non-Math pairs.
(d) 8: Option (d) is incorrect because it overcounts by including invalid combinations where Mathematics II is chosen without Mathematics I.

Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2015, held on 23 August 2015. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.