Goulden Jackson

$$Exact(t) = AtLeast(t-1)$$

Card decks
there are 13 + 4 sorts. There are "exact terms" only for k=2.

$$ E(x) = 52x^2 $$

$$ N(x) = 52 + 102x + 52x^2 $$

the 102 "at least" terms are four aces + four kings + ... + thirteen spades + ... + thirteen clubs.

Red and blue balls
3 red balls and 7 blue balls. There are three properties : ball, red, blue.

$$ E(x) = 10x^2 $$

$$ N(x) = 10 + 20x + 10x^2 $$

https://math.stackexchange.com/questions/2939749/calculate-probability-using-inclusion-exclusion-and-deduce-formula-for-binomial calculate using inclusion-exclusion the probability that m special people are in the group

https://math.stackexchange.com/questions/2854867/calculating-the-probability-of-at-least-one-of-x-i-subset-x-using-inclusion-e

$$\begin{align}

{n-m\choose k-m}=\sum_{j=0}^{m} (-1)^j {m\choose j} {n-j \choose k}

\end{align}$$

https://math.stackexchange.com/questions/2924139/mixture-of-negative-binomial-distributions-technically-some-of-them-are-geometr