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Counting homework, Please explain your answers using full sentences and identifying any of the counting rules that we have discussed so far

Please explain your answers using full sentences and identifying any of the counting rules that we have discussed so far.
1. a) How many compositions of k into n parts exist?
b) How many compositions of k exist (into any number of parts)?
c) If your answer to part (b) included a summation, use a result that we have seen before to write this summation in a closed form (without a summation). 

2. What is the number of compositions of 50 into four odd parts? (Hint: Put one ball in each box and then glue the remaining ones in pairs. This will ensure an odd number of balls in each box.) 

3. Determine p(5) by listing all of the possibilities. 

4. Determine p2(7) by listing all of the possibilities. 

5. What is the number of weak compositions of 10 into four parts, all of which are smaller than nine? (Hint: Count all weak compositions of 10 into four parts. Then subtract off the ones you don’t want.) 

6. We want to divide 10 children among 3 playgrounds. All of the playgrounds are the same size and have the same jungle gym. Not all of the playgrounds need to be used. In how many ways can we do this? (Your answer does not need to be a closed form.) 

7. A community volunteer group of 20 people are organizing a City Park Clean Up day. They need to split themselves into smaller groups each assigned to one of the four city parks. All of the parks are the same size and need the same amount of clean-up. In how many ways can these group assignments be made. Keep in mind that every one of the parks must get cleaned. (Your answer does not need to be a closed form.) 

8. Determine S(4,2) by listing all of the possibilities. 


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