Answers to the problem ‘Paul Erdős International Mathematical Challenge’

Threads:

Eight swimmers competed in an 8-lane swimming pool numbered 1, 2, 3, 4, 5, 6, 7, 8. We know the following about the results of the competition:

i) No one finishes in the same numbered position as his or her lane, and no two finish at the same time.

ii) Athletes in the even lane finish in an even position, athletes in the odd lane finish in an odd position.

Ask how many different finish possibilities are there for the eight runners.

Answer:

Consider the four athletes who initially finished in odd positions 1, 3, 5, 7 whose ordinal number is a permutation of the set (1, 3, 5, 7) with the numbers indicating the starting position. and different destination numbers. Then there are 3 ways to choose the number 1 position later, which is different from the first position 1.

For each way of choosing position 1, there are always 3 ways to choose three numbers 3, 5, 7 later that are different from its original position, for example, if position 1 is in second place, then we have the three sets of numbers are (3, 1, 7, 5); (5, 1, 7, 3); (7, 1, 3, 5). Thus, the number of possibilities for the four athletes to finish in odd positions 1, 3, 5, 7 initially with ordinal numbers is a permutation of the set of numbers (1, 3, 5, 7) with the numbers indicating The difference between starting position and finishing number is 3 x 3 = 9 (way).

Similarly, we have the number of possibilities for the four athletes to finish at first in even positions 2, 4, 6, 8 whose ordinal number is a permutation of the set of numbers (2, 4, 6, 8) with the The difference between the number of starting positions and the number of finishing positions is 3 x 3 = 9 (way).

So the number of ways to the finish line of eight runners is: 9 x 9 = 81 (way).

Tran Phuong