Question

2. How many:
   1. ways can an exam consisting of 10 true and false questions be answered?
   2. ways can an exam consisting of 5 true and false questions followed by 2 multiple choice questions with four choices each be answered?
   3. 4-digit numbers are there?
   4. New York license plates can be made that consist of 3 letters followed by 4 numbers?
   5. 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’?
   6. 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’?
   7. ways can a vase be filled with 5 flowers given that you have 8 flowers?
   8. ways can you seat 4 couples in a concert row?
   1. ways can you seat 4 couples in a concert row given that the couples want to sit with their partners?
   10. ways can seat 4 couples at a circular table given that the couples want to sit with their partners?
   11. 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

= 2¹⁰

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

= 2⁵ * 4² = 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

= 26³ * 10⁴

= 175760000

Therefore there are 175760000 possible license plates here.