Question

part 1 of 2
how many ways can a student pick five questions from an exam containing eleven questions?
there are 462 ways to pick five questions from an exam containing eleven questions.
part: 1 / 2
part 2 of 2
how many ways are there if he is required to answer the first question and the last question?
there are ways if he is required to answer the first question and the last question.

Explanation:

Step1: Calculate the number of ways to pick five questions from eleven

The formula for combinations is \(C(n,r)=\frac{n!}{r!(n - r)!}\), where \(n = 11\) and \(r=5\).

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Step2: Calculate the number of ways when first and last questions are required

If the first and last questions are required, we only need to choose \(5-2 = 3\) questions from the remaining \(11 - 2=9\) questions.
Using the combination formula \(C(n,r)\) with \(n = 9\) and \(r = 3\)

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Answer:

Part 1: \(462\) ways.
Part 2: \(84\) ways.