phxmba Posted March 1, 2003 Share Posted March 1, 2003 I saw this in one of the websites 1. in how many ways can you seat 8 students in a row of 8 chairs ? answer: 8! 2. in how many ways can you seat 8 students in a row of 9 chairs ? answer: 9! 3. in how many ways can you seat 8 students in a row of 10 chairs ? answer: 10! / 2! 4. in how many ways can you seat 8 students in a ring of 8 chairs ? answer: 7! I wasn't sure about the answer to questions 2 & 3 - how they are different. Can someone explain? Quote Link to comment Share on other sites More sharing options...
phase Posted March 2, 2003 Share Posted March 2, 2003 You should take into account that seats without poeple are the same. If you don't consider this,the answer is 10!,but when you change the two free seats,you can figure out that the array isn't changed,so 10!/2! Furthermore,in the problem below, in how many ways can you seat 8 students in a row of 11 chairs ? the answer is 11!/3! clear? Quote Link to comment Share on other sites More sharing options...
phxmba Posted March 2, 2003 Author Share Posted March 2, 2003 Hi Phase, I am sorry. It is still not very clear to me as to why we need to divide by 2!. Can you please elaborate a bit? Thanks Quote Link to comment Share on other sites More sharing options...
phase Posted March 2, 2003 Share Posted March 2, 2003 I will interpret it in such a way: Assuming that there is an apple on one free seat , we can change this problem to "in how many ways can you seat 10 students in a row of 10 chairs ? ",thus the answer is 10!.Then,I take away the apple,the situation changes into "in how many ways can you seat 8 students in a row of 10 chairs ? ".We assume there are X ways. I put the former apple on one of the free seats.I have 2! chooses.So I can obtain an equation:2!*X=10!,X=10!/2!. Clear?Come on. Quote Link to comment Share on other sites More sharing options...
phxmba Posted March 3, 2003 Author Share Posted March 3, 2003 Thanks Phase. Also, I got another explanation to the problem. Ways of seating 8 students in 8 seats - 8P8 -> 8! Ways of seating 8 students in 9 seats (selecting 8 out of available 9) - 9P8 -> 9!/1! Ways of seating 8 students in 10 seats (selecting 8 out of available 10) - 10P8 -> 10!/2! Quote Link to comment Share on other sites More sharing options...
phase Posted March 4, 2003 Share Posted March 4, 2003 Congratulations ,you got it and gave a better solution. Quote Link to comment Share on other sites More sharing options...
Recommended Posts
Join the conversation
You can post now and register later. If you have an account, sign in now to post with your account.