A conference for budding entrepreneurs is being conducted at an engineering college.
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A conference for budding entrepreneurs is being conducted at an engineering college. A total of 200 students of varying age are attending this conference.
In the given scenario, which of these statements will help you determine if there are any students in the conference who are of the same age and from the same engineering college?
Statement 1: The range of ages of the students is 18 to 22, inclusive.
Statement 2: Students from around 15 engineering colleges are attending this conference.
Show answer & explanation
With ages 18-22 (5 ages) and 15 colleges, there are 5 * 15 = 75 distinct (age, college) pairs. By Pigeonhole Principle, among 200 students, at least ceil(200/75) = 3 students must share the same age and college.
Step-by-step Derivation:
Step 1: Statement 1 gives 5 possible age values: {18, 19, 20, 21, 22}.
Step 2: Statement 2 gives 15 colleges.
Step 3: Total possible combinations of (age, college) = 5 * 15 = 75 pigeonholes.
Step 4: With 200 students distributed into 75 categories, by Pigeonhole Principle at least ceil(200/75) = 3 students must share the exact same age and college.
Step 5: Neither statement alone provides the number of pigeonholes. Both together are sufficient.