The reason this is so complicated is because of a mathematical principle called Factorials.

它之所如此复杂，因为阶乘一数学原理。

Instead of factorial time, it takes linear time.

它不阶乘时间， 而线性时间。

So the n minus k will become b minus a factorial.

所 n 减去 k 将变成 b 减去一个阶乘。

If this was 7 over 7 minus 2 factorial we would have 7 times 6.

如 7 除 7 减 2 阶乘，们将有 7 乘 6。

For those who remember from math class, that is seven factorial.

那些记得数学课上的人来说，七个阶乘。

So times lambda to the k k over k factorial.

所乘 lambda 到 k k 阶乘。

So you end up with 1 times lambda k over k factorial.

所你最终得到 lambda k 的 1 倍 k 的阶乘。

Divided by 2 factorial times e to the minus 9 power.

除 2 阶乘 e 的负 9 次方。

And as you'll see it's actually 2 factorial ways that it can happen.

正如您将看到的，它实际上可通过两种阶乘方式发生。

You may have heard that there is a function generalizing the factorial to real and even complex inputs, the gamma function.

您可能听说过有一个函数将阶乘推广到实数甚至复数输入，即伽马函数。

This video by Michael DiFranco about extending the factorial offers another great example of a lesson with good motivation along the way.

Michael DiFranco 制作的段关扩展阶乘的视频提供了另一个很好的例子，说明了一路上具有良好动机的课程。

If this had 3 we would do 3 factorial, and I'll show you how that can happen.

如它有 3，们会做 3 个阶乘，会告诉你如何发生的。

5 factorial is 5 times 4 times 3 times 2 times 1.

5阶乘5乘4乘3乘2乘1。

Once again, you'll have to know that 0 factorial is equal to 1.

再一次，你必须知道 0 的阶乘等 1。

So k factorial could be written as k times k minus 1 factorial.

所 k 阶乘可写成 k 乘 k 减 1 阶乘。

And here we can make a little bit of a simplification because what's k divided by k factorial?

在里们可稍微简化一下，因为 k 除 k 的阶乘多少？

And we just showed that this is equal to lambda to the kth power over k factorial times e to the minus lambda.

们刚刚证明等 lambda 的 k 次方乘 e 的 k 阶乘乘负 lambda。

So this could be rewritten as k times k minus 1 factorial.

所可改写为 k 乘 k 减 1 阶乘。

So we could rewrite n factorial using the same trick up here.

所们可在里使用相同的技巧重写 n 阶乘。

To account for that, you divide out by the extent  to which you've overcounted, the number of permutations of four  items which looks like four factorial.

为了解释一点， 您可除多算的程度，即四个项目的排列数量， 看起来像四个阶乘。