1)A person has two sons and two daughters. There are three boys' schools and two girls' schools in the area where they lives.In how many different ways can the children be sent to school?In how many ways can the children be sent to school such that no two children attend the same school?
Choose a school for the 1st boy in 3 ways.
For each of those 3 ways to place the first boy in a school,
you can choose a school for the 2nd boy in 2 ways.
That's 3*2 = 6 ways to send the 2 boys to different schools.
For each of those 6 ways to place the two boys in different school,
you can choose a school for the 1st girl in 2 ways.
That's 6*2 = 12 ways to send the boys and the 1st girl to different
schools.
For each of those 12 ways to place the boys and the 1st girl, there
will be only 1 girl left, and only 1 school left, so there'll just
be 1 way to send her, namely to that 1 remaining school.
Answer 3*2*2*1 = 12 ways
2)A shirt factory produces shirts of three different sizes. It is expected to produce shirts with two different materials and four different colors for the festive season. How many different shirts can be produced?
Choose the material 2 ways.
Choose the color 4 ways.
Answer 2*4 = 8 ways.
3)Find the number of all positive numbers with not more than four digits that can be made using the digits 0,1,2,3,4 and 5.
We can consider numbers less than 1000 to be 4-digit numbers by beginning
them with 0's. For instance 344 can be written as a 4 digit number 0344,
and 2 can be wrotten as a 4-digit number 0002.
Answer:
Choose the 1st digit 6 ways.
Choose the 2nd digit 6 ways.
Choose the 3rd digit 6 ways.
Choose the 4th digit 6 ways.
That's 6*6*6*6 = 64 = 1296 ways
However that includes the number 0000, which is not positive.
So we must subtract 1:
Answer: 1295.
Find also the number of numbers out of these in which each digit is not repeated.
We cannot consider numbers less than 1000 to begin with 0's because for
instance 0002 would have repeated 0's.
1 digit numbers.
There are 5 of them: 1,2,3,4 and 5
That's 5 1-digit numbers.
2 digit-numbers.
Choose the first digit 5 ways, 1,2,3,4 or 5. (can't be 0)
Choose the second digit 5 ways, (can be 0).
That's 5*5 = 25 2-digit numbers.
3 digit-numbers.
Choose the first digit 5 ways, 1,2,3,4 or 5. (can't be 0)
Choose the second digit 5 ways, (can be 0).
Choose the third digit 4 ways,
That's 5*5*4 = 100 3-digit numbers.
4 digit-numbers.
Choose the first digit 5 ways, 1,2,3,4 or 5. (can't be 0)
Choose the second digit 5 ways, (can be 0).
Choose the third digit 4 ways,
Choose the fourth digit 3 ways,
That's 5*5*4*3 = 300 4-digit numbers.
Grand total + 5+25+100+300 = 430
Edwin