Question

At a university all students must take a test to see if they understand campus polices. The test has 6 multiple choice questions, each with three possible answers and only one right answer per question. This year every student got at least two questions right. How many possible ways could the students have answered the test questions?

The incoming class this year has 531 students. Prove that at least two students answered the test in the exact same way.

**Please Explain**

Answer #1

At a university all students must take a test to see if they
understand campus polices. The test has 6 multiple choice
questions, each with three possible answers and only one right
answer per question. This year every student got at least two
questions right. How many possible ways could the students have
answered the test questions?
The incoming class this year has 541 students. Prove that at
least two students answered the test in the exact same way.

1. There are 65 students, 35 females and 30 males in a class.
How many ways to choose 3 students so that 1 of them is male?
2. A student is writing a multiple choice test with 20
questions. Each question has 4 possible answers. The student will
randomly choose one of the 4 answers for each question. What is the
probability the student answers at least 2 questions correctly?
(please answer questions step by step)

A student guesses on every question of a multiple-choice test
that has 7 questions, each with 3 possible answers. What is the
probability that the student will get at least
5 of the questions right?

Please answer this with as much detail as possible so I can
better understand it.
Thank you in advance!
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