discrete math prob
1- An office building contains 27 floors and has 37 offices on each floor. How many offices are in the building?
Solution:
Total number of offices is 27 * 37 =999
2- A particular brand of shirt comes in 12 colors, has a male version and a female version, andcomes in three sizes for each sex. How many different types of this shirt are made?
Solution:
Using product rule, we have 12×2×3 = 72 different types of this shirt.
3. Use a tree diagram to find the number of bit strings of length four with no three consecutive 0s.
Sol:
4. How many ways are there to arrange the letters a,b,c,and d such that a is not followed immediately by b?
With "a" in the 1st position: acbd acdb adbc adcb
With "a" in the 2nd position: bacd badc cadb dacb
With "a" in the 3rd position: bcad bdac cbad dbac
With "a" in the 4th position: bcda bdca cbda cdba dbca dcba
There are 18 ways
5. How many different plates can be made with one letter followed by one number?
Sol:
There are 26 choices of letters. Each of these 26 letters can be followed by one of 10
Different digits--0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
Therefore there are 26 x1 0 = 260 different plates which can be made 6. How many license plates can be made using either two or three letters followed by either two or three digits?
Sol:
For two letters, there are 26 choices for the first letter and 26 for the second. For the two letters, there are 10 choices for the first letter and 10 for the second.
That's 26²(10²) = 67600
For two letters followed by three numbers, multiply by 10 again:
26²(10²)(10) = 676000
For three letters, 26²(26)
With two numbers, 26²(26)(10²) = 1757600
With three numbers, 26²(26)(10²)(10) = 17576000
Adding them up
67600 + 676000 + 1757600 + 17576000 = 20077200
Solution:
Total number of offices is 27 * 37 =999
2- A particular brand of shirt comes in 12 colors, has a male version and a female version, andcomes in three sizes for each sex. How many different types of this shirt are made?
Solution:
Using product rule, we have 12×2×3 = 72 different types of this shirt.
3. Use a tree diagram to find the number of bit strings of length four with no three consecutive 0s.
Sol:
4. How many ways are there to arrange the letters a,b,c,and d such that a is not followed immediately by b?
With "a" in the 1st position: acbd acdb adbc adcb
With "a" in the 2nd position: bacd badc cadb dacb
With "a" in the 3rd position: bcad bdac cbad dbac
With "a" in the 4th position: bcda bdca cbda cdba dbca dcba
There are 18 ways
5. How many different plates can be made with one letter followed by one number?
Sol:
There are 26 choices of letters. Each of these 26 letters can be followed by one of 10
Different digits--0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
Therefore there are 26 x1 0 = 260 different plates which can be made 6. How many license plates can be made using either two or three letters followed by either two or three digits?
Sol:
For two letters, there are 26 choices for the first letter and 26 for the second. For the two letters, there are 10 choices for the first letter and 10 for the second.
That's 26²(10²) = 67600
For two letters followed by three numbers, multiply by 10 again:
26²(10²)(10) = 676000
For three letters, 26²(26)
With two numbers, 26²(26)(10²) = 1757600
With three numbers, 26²(26)(10²)(10) = 17576000
Adding them up
67600 + 676000 + 1757600 + 17576000 = 20077200