题目253：整理拼图

欧拉计划

Tidying up

A small child has a “number caterpillar” consisting of forty jigsaw pieces, each with one number on it, which, when connected together in a line, reveal the numbers 1 to 40 in order.

Every night, the child's father has to pick up the pieces of the caterpillar that have been scattered across the play room. He picks up the pieces at random and places them in the correct order.
As the caterpillar is built up in this way, it forms distinct segments that gradually merge together.
The number of segments starts at zero (no pieces placed), generally increases up to about eleven or twelve, then tends to drop again before finishing at a single segment (all pieces placed).

For example:

Piece Placed	Segments So Far
12	1
4	2
29	3
6	4
34	5
5	4
35	4
…	…

      
Let M be the maximum number of segments encountered during a random tidy-up of the caterpillar.
For a caterpillar of ten pieces, the number of possibilities for each M is

M	Possibilities
1	512
2	250912
3	1815264
4	1418112
5	144000

              
so the most likely value of M is 3 and the average value is ³⁸⁵⁶⁴³⁄₁₁₃₄₀₀ = 3.400732, rounded to six decimal places.

The most likely value of M for a forty-piece caterpillar is 11; but what is the average value of M?

Give your answer rounded to six decimal places.

题目：

一个小孩有一个由 40 个拼图块构成的＂数字毛毛虫＂，每个拼图块上有一个数字。当拼图块连接成一条线时，将按顺序显示 1 到 40。

每天晚上，小孩的父亲不得不去收拾散落在各个房间的毛毛虫拼图块。他随机地捡起并将拼图块按正确的顺序放置。

按这种方式，先得到不同的拼图段，渐渐地拼图段连接起来，最后形成完整的毛毛虫拼图。

拼图段的数目从 0 开始（此时没有拼图块放置），渐渐地这个数目增加到 11 或 12，在一个完整的拼图即将完成之前，又趋向减少。（所有拼图块都已放置）


例如：

放置的拼图块数目	目前可能的拼图段数目
12	1
4	2
29	3
6	4
34	5
5	4
35	4
…	…

      
设 M 为随机整理毛毛虫拼图时出现的最大拼图段数。

对于一个由 10 个拼图块构成的毛毛虫拼图，对于M的每种取值，相应的情况数目为：

M	相应的情况数目
1	512
2	250912
3	1815264
4	1418112
5	144000

所以 M 最可能的值为 3，平均值为 ^{(1*512+2*250912+3*1815264+4*1418112+5*144000)}⁄_10! = 3.400732，四舍五入到小数点第六位。

对于一个由 40 个拼图块构成的毛毛虫拼图 M 最可能的值为 11，但 M 的平均值为多少？

给出你的答案，四舍五入到小数点第六位。