Question

- How many:
- ways can an exam consisting of 10 true and false questions be answered?
- ways can an exam consisting of 5 true and false questions followed by 2 multiple choice questions with four choices each be answered?
- 4-digit numbers are there?
- New York license plates can be made that consist of 3 letters followed by 4 numbers?
- New York license plates can be made that consist of 3 letters followed by 4 numbers given the restriction that if the first letter is an ‘N’, then the second letter must be a ‘Y’?
- New York license plates can be made that consist of 3 letters followed by 4 numbers given the restrictions that (1) if the first letter is an ‘N’, then the second letter must be a ‘Y’ and (2) if the second letter is an ‘N’, then the third letter must be a ‘Y’?
- ways can a vase be filled with 5 flowers given that you have 8 flowers?
- ways can you seat 4 couples in a concert row?
- ways can you seat 4 couples in a concert row given that the couples want to sit with their partners?
- ways can seat 4 couples at a circular table given that the couples want to sit with their partners?
- ways can a necklace be strung together with 24 different beads? (Note: this is different than a standard circular problem.)

Answer #1

We would be looking at the first 4 parts here as:

a) Number of ways to attempt 10 True and False questions is computed here as:

= (Number of ways to attempt a True False Question
)^{10}

= 2^{10}

**Therefore there are 1024 ways to answer the 10 questions
here.**

b) Number of ways to attempt exam here is computed as:

= Number of ways to attempt 5 True False questions * Number of ways
to attempt 2 Multiple choice questions

= 2^{5} * 4^{2} = 512

**Therefore there are 512 ways to answer the questions
here.**

c) Number of 4 digit numbers is computed here as:

= 9999 - 999 = 9000

**Therefore there are 9000 four digit numbers
here.**

d) Number of ways to form the license plate here is computed as:

= (Total number of letters possible )^{3} * (Number of
digits possible )^{4}

= 26^{3} * 10^{4}

= 175760000

**Therefore there are 175760000 possible license plates
here.**

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