114. Division by 11. If the nine digits $1,2,3,4,5,6,7,8,9$ are written in a random order, for example 412539768, what is the probability that the resulting number is divisible by 11? The number I wrote, of course, is not divisible by 11, but if you swap the 1 and 8 in it, it will be divisible by 11.
114. For a number to be divisible by 11, it is necessary that either four alternating digits sum to 17, and the remaining five sum to 28, or vice versa. For example, in the given number (482539761), the digits \(4,2,3\), 7,1 sum to 17, and \(8,5,9,6\) sum to 28. Furthermore, four digits can sum to 17 in nine different ways, and five digits can sum to 17 in two ways. This results in a total of 11 ways. In each of these 11 cases, the four digits can be permuted in 24 ways, and the five digits in 120 ways, giving 2880 variations. In total, the number of favorable outcomes is \(2880 \times 11 = 31680\).

Since nine digits can be permuted in 362880 ways, we get 115 to 11 against the randomly chosen number being divisible by \(11^{*}\).