Total students in Arts, Commerce, and Science — which statements suffice? I. 20% of the students study Science. II. Numbers studying Arts and Commerce are in the ratio 3 : 5. III. Commerce exceeds Science by 375 students.

Introduction / Context:We must find which statements are sufficient to determine the total number of students across three streams (Arts, Commerce, Science).

Given Data / Assumptions:

• I: Science S = 0.20 * T, where T is total students.
• II: Arts : Commerce = 3 : 5 ⇒ A = (3/5) * C.
• III: Commerce exceeds Science by 375 ⇒ C = S + 375.

Concept / Approach:We need enough independent relations to solve for T. Three unknown categories (A, C, S) with total T require either two independent equations plus T = A + C + S, or equivalent information to eliminate variables.

Step-by-Step Solution:

Verification / Alternative check:With T known, all categories are determined uniquely (S = 250, C = 625, A = 375). Any pair of statements without the third leaves at least one variable free.

Why Other Options Are Wrong:

• II and III only: Lacks direct link from S to T.
• III and either I or II only: III + I lacks the A–C relation; III + II lacks S–T relation.
• Any two of the three: As shown, two are insufficient.

Common Pitfalls:Using the ratio to equate A and C incorrectly; forgetting to include T = A + C + S in the system.

Final Answer:All I, II and III