Data Sufficiency — Speeches over a two-day programme: How many total speeches were delivered in the two days? Statements: I. 18 speakers were invited to give at least one speech (maximum two each), but one-sixth of the speakers could not come. II. One-third of the speakers delivered two speeches each.

Introduction / Context:
This is a classic data sufficiency problem. We must decide whether each statement (individually or together) gives enough information to compute the exact number of speeches delivered over two days, without necessarily calculating the numerical value unless required for sufficiency.

Given Data / Assumptions:

• Each invited speaker can deliver a minimum of 1 and a maximum of 2 speeches.
• Statement I: 18 invited; one-sixth absent.
• Statement II: One-third of the speakers (who attended) gave 2 speeches each.
• We interpret “speakers” in Statement II as those who actually attended and spoke.

Concept / Approach:
Compute the number of attendees from Statement I, then use Statement II to split attendees into two groups: those who delivered 2 speeches and those who delivered exactly 1 speech. Sum to get total speeches. Check sufficiency of each statement alone and then together.

Step-by-Step Solution:
Absent speakers = (1/6) * 18 = 3.Attendees = 18 − 3 = 15.From Statement II: Speakers giving two speeches = (1/3) * 15 = 5.Speakers giving one speech = 15 − 5 = 10.Total speeches = (5 * 2) + (10 * 1) = 10 + 10 = 20.

Verification / Alternative check:
Statement I alone leaves ambiguity because we do not know how many selected the 2-speech option. Statement II alone lacks the base number of attendees. Only together do they yield a unique total of 20.

Why Other Options Are Wrong:

• I alone: Insufficient—missing the 1-vs-2 split.
• II alone: Insufficient—missing the total attendees.
• Either alone: False; each lacks complementary information.
• Neither alone: False because together they are sufficient.

Common Pitfalls:

• Assuming “one-third” applies to the invited (18) instead of the attendees (15).
• Forgetting to account for absentees before applying the two-speech proportion.

Final Answer:
Both I and II are sufficient