12. (20 points) After a very successfiul surting season, the Interstellar Surfin

Question

12. (20 points) After a very successfiul surting season, the Interstellar Surfing Association (ISA) is putting on surfing demonstrations. There are 8 demonstrations total, which are split up between the two champion surfers (Alex and Tony, of course). Neither surfer can perform 4 demonstrations in a row, but either of them can perform more (or less) than 4 demonstrations total. For example, one possible arrangement is for Alex to do 2 demonstrations, then Tony do 3, then Alex do 3. Determine how many possible ways are there for the ISA to organize those 8 demonstrations. You do not need to simplify your answer.

Explanation / Answer

There are 8 demonstrations and we have two surfers Alex and Tony. It can be understand as we have 8 places to fill and having 2 choices for each place. Therefore 8 demonstrations can be perfomaned by 28 number of ways. Given that each surfer can not perforn 4 and more demonstrations in a row but his total demonstrations can be more than 4 (or less). Then the number of possible ways so that ISA can organize those 8 demonstrations

= 28 - (# of ways in which each surfer can perform 4 or more demonstrations in a row)

Now our job is to calculate the number of ways in which each surfer can perform 4 or more demonstrations in a row. Thus, if exactly 4 perfomances appear in a row then the possible ways of arrangements are

{ AAAATAAA, AAAATAAT, AAAATATA, AAAATTAA, AAAATTTA, AAAATTAT, AAAATATT, AAAATTTT, TAAAATAA, TAAAATAT, TAAAATTA, TAAAATTT, TTAAAATA, ATAAAATT, TTAAAATT, TTTAAAAT, ATTAAAAT, TATAAAAT,AATAAAAT,TTTTAAAA, TTATAAAA, TATTAAAA,ATTTAAAA, AATTAAAA,TAATAAAA, ATATAAAA, AAATAAAA } and same arrangement can be done for exactly four T's threfore 54 all possible no.of ways.

  If exactly 5 perfomances appear in a row then the possible ways of arrangements are

{AAAAATAA, AAAAATTA, AAAAATAT, AAAAATTT, TAAAAATA, TAAAAATT, TTAAAAAT, ATAAAAAT, TTTAAAAA, TATAAAAA, ATTAAAAA, AATAAAAA} and same arrangement can be done for exactly five T's threfore 24 all possible no.of ways.

If exactly 6 perfomances appear in a row then the possible ways of arrangements are

{AAAAAATA, AAAAAATT, TAAAAAAT, TTAAAAAA, ATAAAAAA} and same arrangement can be done for exactly six T's threfore 10 all possible no.of ways.

If exactly 7 perfomances appear in a row then the possible ways of arrangements are

{AAAAAAAT, TAAAAAAA, AAAAAAAA} and same arrangement can be done for exactly seven T's threfore 6 all possible no.of ways.

Therefore the number of ways in which each surfer can perform 4 or more demonstrations in a row

= 54+ 24+ 10+ 6= 94

Hence, the number of possible ways so that ISA can organize those 8 demonstrations

= 28 - 94= 256- 94= 162

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