The entrance test to Podunk University consists of 6 TrueFal

The entrance test to Podunk University consists of 6 True-False questions. A subtest consists of two of the True-False questions. Explain why there are 15 subtests. Explain why no matter how one answers the 6 True-False questions, there are at least 6 subtests that you either got 0 or 2 correct. Use the strong pigeonhole principle to explain why if 60 people take the exam, then there is a subtest in which at least twelve people answered both questions correctly, or a subtest in which at least twelve people answered both questions incorrectly.

Solution

Ans(a):

two subtests can be selected from 6 tests in C(6,2) ways where C indicates Combination.

C(6,2)=6!/[2!*4!]=6*5*4!/[2*4!]=6*5/2=15

That\'s why there are 15 subtests.