The first "question" is a list of these formulas:
¬∀x P(x) is the same as ∃x ¬P(x). Same if you reverse ∀ and ∃.
(n choose k) = n! / (n-k)!k! = n×(n-1)×...×(n-k+1) / k×(k-1)×...×3×2×1
Counting overlapping sets: |A∪B| = |A| + |B| - |A∩B|
Summing an arithmetic series: reverse the order and add by column. All columns will be the same.
Summing a geometric series with ratio r: calculate rS - S, where S is the sum.
Stars and Bars: when making k choices spread
over n categories (k cookies to n children, or k scoops of ice cream
from n flavors, or alphabetically ordered strings of letters of length
k, from among n letters)
Use k stars, and n-1 bars.
a|b: a divides b: ∃k ak = b
Modular arithmetic mod n: divide by n and take the remainder.
Fermat's little theorem: if p is prime, then ap ≡ 1 mod p for all a ≢ 0
Bayesian vs Frequentist understanding of probability