A Teacher puts together a test worth 100 points. There are 10 questions. Each question must award atleast a score of 5 points. Each score is a whole number. How many ways are there to arrange the 100 points?
Asked
Active
Viewed 32 times
-2
-
1What are the allowable scores for a question? Can a question be worth $5+\sqrt 2$ points? How about $ 7? 43?$ Does order matter? Is the first five being worth $15$ and the last five being worth $5$ different from starting with five questions worth $5$? Please think about your question and supply enough information to answer it. – Ross Millikan Apr 10 '17 at 20:31
-
1The only constraint is that each individual score cannot be lower than 5. Everything else goes. So 50 of those points have effectively already been set. – Grimchester Apr 10 '17 at 20:35
-
Then there are continuum many ways to arrange the points. -1 – Ross Millikan Apr 10 '17 at 20:36
-
What if the only allowed points are whole numbers. – Grimchester Apr 10 '17 at 20:37
-
changed description to reflect constraints properly. – Grimchester Apr 10 '17 at 20:39
-
Hint: stars and bars – hardmath Apr 11 '17 at 01:30
1 Answers
1
Once you assign the mandatory $5$ points to each question, you have $50$ left to distribute. Assuming the order matters, you are looking for weak compositions of $50$ into $10$ parts which can be computed by the usual stars and bars approach. This gives ${59 \choose 9}=12\ 565\ 671\ 261$ compositions
Ross Millikan
- 374,822