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

它之所以如此复杂，是因为阶乘这一数原理。

Instead of factorial time, it takes linear time.

它不是阶乘时间， 而是线性时间。

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

所以 n 减去 k 将变成 b 减去一个阶乘。

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

如果这是 7 除 7 减 2 阶乘，们将有 7 乘以 6。

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

对于那些记得数课上说，这是七个阶乘。

So times lambda to the k k over k factorial.

所以乘以 lambda 到 k k 阶乘。

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

所以你最终得到 lambda k 1 倍于 k 阶乘。

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

除以 2 阶乘 e 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，它实际上以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 制作这段关于扩展阶乘视频提供了另一个很好例子，说明了一路上具有良好动机课程。

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 个阶乘，会告诉你这是如何发生。

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

5阶乘是5乘以4乘以3乘以2乘以1。

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

再一次，你必须知道 0 阶乘等于 1。

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

所以 k 阶乘以写成 k 乘以 k 减 1 阶乘。

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

在这里们以稍微简化一下，因为 k 除以 k 阶乘是多少？

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 k 次方乘以 e k 阶乘乘以负 lambda。

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

所以这以改写为 k 乘 k 减 1 阶乘。

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

所以们以在这里使用相同技巧重写 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.

为了解释这一点， 您以除以多算程度，即四个项目排列数量， 看起像四个阶乘。