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

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!