Question

# How many: ways can an exam consisting of 10 true and false questions be answered? ways...

1. 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?
9. 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.)

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.

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