Let A={a,b,c,d} 1. Find all combinatorial lines in A^2. How many combinatorial lines are there? 2. Let n in NN. Prove that the number of combinatorial lines in A^n equals 5^n-4^n

nicekikah 2020-10-20 Answered
Let A={a,b,c,d}
1. Find all combinatorial lines in A2. How many combinatorial lines are there?
2. Let n in N. Prove that the number of combinatorial lines in An equals 5n4n
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Expert Answer

SkladanH
Answered 2020-10-21 Author has 80 answers

1) For any lA , let us define
B1l={(l,x):xA}
B2l={(x,l):xA}
C={(x,x):xA}
These above mentioned are the three combinatorial lines in A2. There is only one combinatorial line of type C , which is C itself. |A| = 4, there are four combinatorial lines B1, ,t in A and there are four combinatorial lines in A2.These are all the combinatorial lines in A2. Hence there are a total of 9 combinatorial lines in A2.
2) Consider any Subset E{1,.n},E . For every i{1,n} E , fix any ai in A.
Then we get a combinatorial line defined as:
BE,{ai}IiE={x1,..xn}An:(xi=aIiE)and(xi=xiiE)and(xA)}
We further note that reach combinatorial line is of the form , since there is only one free coordinate in a combinatorial line.
Now , for any F{1,n} , the number of subsets of {1.n} of size F is equal to (nk)
Further , for any subset E{1,.n},|E|=F , the number of choices for the remaining coordinates i{1,2.n}
iE is equal to |A|n|E|=|A|nF=4nF
Hence the total number of combinatorial is equal to :
F=1n=(F=0n(nk)4nF)4n
=(F=0n(nk)4F)4n
=(4+1)n4n
=5n4n
By Binomial Theroem .

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