S = {1, 2, …, 100}. How many permutations are there of S in which the number 1 is next to at least one even number?

Question

S = {1, 2, …, 100}. How many permutations are there of S in which the number 1 is next to at least one even number?

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Ayla 1 month 2021-09-16T02:45:21+00:00 1 Answer 0

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    2021-09-16T02:47:14+00:00

    Answer:

    Check the explanation

    Step-by-step explanation:

    in this particular type of problems,we have to go with the cases.we know there would be a 50 odd-even pairs between 1 and 100.

    1st case: 1 is placed at the head position of given set S.so there is a chance of only one even number beside 1.so there are 50×98!

    2nd case: 1 is placed at tail position of S.so there is again a chance of one even number beside 1. so there are

    50×98!

    3rd case: 1 is placed neither at head nor at tail.1 can be at rest 98 places.lets fix the place of 1.then the choices are difference between permuting all the numbers and 1 surrounded by two odd numbers.

    i.e., 99! – 49p2 ×97!

    so total choces are = 2×50×98! + 99! – 49p2 × 97!

    = 7448 × 97! + 99!

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