How can i calculate the result from this formula?

Question

There is a question posted here that the "Generalized Inclusion-Exclusion Principle" is used to be answered. See here: Probability of number of unique numbers in $37$ Roulette Wheel spins.

In this type: $$\sum_{k=0}^{37}(-1)^{k+m-37}\binom{k}{37-m}\binom{37}{k}(37-k)^n$$

Plugging in m=24 and n=37 we get the result:

2157142399433325078661979618737564774321235951616000000000

My actual question is can someone explain me how plugging these 2 we get to that result? Also what number should be plugged in the "k" variable? It is not mentioned so i don't know if something must be plugged in k to get this result.

thanks everyone for any answer in advance!

Answer 1

The $k$ variable is declared inside the summation. The summation tells us to take the expression and evaluate it for $k=0, 1, 2, ... 37$. Then, we take all of those results and add them up to get our answer. Thus, for $m=24$ and $n=37$, we evaluate this expression: $$(-1)^{k+24-37}\binom{k}{37-24}\binom{37}{k}(37-k)^{37}$$ for all $k=0, 1, 2, ... 37$ and then sum the result of all of those expressions to get our answer.

Answer 2

This calculation should never be done by hand, so I can only assume that they used s computer. Likewise, I certainly cannot do out the calculation to produce that number.

The reason why there is no number being plugged in for $k$ is because $k$ is not a variable. It's the index of summation. Are you familiar with the expression $\sum_{k=0}^n f(k)$?