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  • Directions for question Under the reign of the Roman emperor Commodus in 600 C.E., ‘munera’s, or gladiator fights, held in the Colo

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Under the reign of the Roman emperor Commodus in 600 C.E., ‘munera’s, or gladiator fights, held in the Colosseum Arena in Rome were a very common occurrence.

The entrance to the Colosseum Arena by the gladiators took place through five underground corridors opening out to five gates with doors arranged in a row. On one day of the gladiator fight, five infamous gladiators – Spartacus, Crixus, Maximus, Flamma and Tetraites – were to enter the arena through the gates, from left to right respectively, in that order. He whose door was opened had to step out on the arena and fight. Who of the gladiators would fight who depended entirely on the fancy of Emperor Commodus, who controlled the opening or closing of doors using a system of codes. More than two gladiators could fight each other in the arena.

If the door to a gate was to be opened, the code was A (A for Aperio in Latin or open), and if the door to a gate was to be closed, the code was C (C for Clausula in Latin or close). Thus if Emperor Commodus finally stopped at the code ACACC, it meant that he wanted the doors of the first and third gate from the left to the right to be opened, that is Spartacus and Maximus should fight.

On that particular day Emperor Commodus started when the code was initially set at CACAC. He could change the code, but only by using some number of steps, wherein in each step the state of only two consecutive doors could be changed. However he had to reach his desired code using the minimum number of steps. At least one change was mandatory.

Question

How many distinct gladiator fights could Emperor Commodus initiate at the Colosseum Arena on that particular day ?

 

Option: 1

16


Option: 2

-


Option: 3

-


Option: 4

-


Answers (1)

best_answer

On that particular day Emperor Commodus started when the code was already set at CACAC, where it can be noted that C and A are in alternate arrangement with three doors closed and two doors opened alternately.

Since in each step of change the code of only two consecutive doors could be changed, so however many may be the number of steps applied, in the final state of the code there can only be an odd number of doors closed

That is the number of C’s present in the final code can be one, three or five, with any gate capable of being coded C, depending upon the number of times the steps were applied by Emperor Commodus 

Similarly, in the final state of the code there can only be an even number of doors opened.

That is the number of A’s present in the final code can be four, two or zero, which also is the complementary result if the number of C’s present in the final code is one, three or five, and hence similarly any gate was capable of being coded A, depending upon the number of times the steps were applied by Emperor Commodus

In the final code the number of ways one C can be present (that is four As present) = 5C1 = 5

In the final code the number of ways three Cs can be present (that is two As present) = 5C3 = 10

In the final code the number of ways five Cs can be present (that is zero As present) = 5C5 = 1

Thus the number of distinct ways in which the final code for the doors of the five gates can be formed = 5+10+1 = 16

Hence, the number of distinct gladiator fights could Emperor Commodus initiate at the Colosseum Arena on that particular day = 16




 

Posted by

shivangi.bhatnagar

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