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

它之所以如此复杂，是因为这一数学原理。

Instead of factorial time, it takes linear time.

它不是时， 而是线性时。

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

所以 n 减去 k 将变成 b 减去一个。

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

如果这是 7 除 7 减 2 ，们将有 7 以 6。

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

对于那些记得数学课上的人来说，这是七个。

So times lambda to the k k over k factorial.

所以以 lambda k k 。

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

所以你最终得 lambda k 的 1 倍于 k 的。

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

除以 2 e 的负 9 次方。

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

正如您将看的，它实际上以通过两种方式发生。

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

您能听说过有一个函数将推广实数甚至复数输入，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 制作的这段关于扩展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.

如果它有 3，们会做 3 个，会告诉你这是如何发生的。

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

5是5以4以3以2以1。

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

再一次，你必须知道 0 的等于 1。

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

所以 k 以写成 k 以 k 减 1 。

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

在这里们以稍微简化一下，因为 k 除以 k 的是多少？

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

们刚刚证明这等于 lambda 的 k 次方以 e 的 k 以负 lambda。

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

所以这以改写为 k k 减 1 。

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

所以们以在这里使用相同的技巧重写 n 。

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.

为了解释这一点， 您以除以多算的程度，即四个项目的排列数量， 看起来像四个。