Darasa la 9–12Discrete MathematicsSomo linapatikana

Combinatorics

Permutations and combinations, binomial coefficients and Pascal's triangle, the binomial theorem, the pigeonhole principle and inclusion–exclusion, with combinatorial proofs.

math.discrete.combinatorics

Fungua somo Onyesha kila kinachohitajika kufikia mada hii

Inajengwa juu ya

Counting principles Proof: direct, contrapositive, contradiction and induction Polynomials

Masharti ya awali ya lazima 15 (hadi chini kabisa)

Utajifunza

Compute binomial coefficients and prove identities such as C(n, k) = C(n − 1, k − 1) + C(n − 1, k) by counting in two ways.
Prove and apply the binomial theorem.
Apply the pigeonhole principle and inclusion–exclusion to counting and existence problems.

Fasili

C(n, k), read 'n choose k', is the number of k-element subsets of an n-element set.
Pigeonhole principle: if more than n objects are placed in n boxes, some box contains at least two objects.

Nadharia

C(n, k) = n!/(k!(n − k)!) for 0 ≤ k ≤ n.
Binomial theorem: (x + y)ⁿ = Σ_{k=0}^{n} C(n, k) xᵏ yⁿ⁻ᵏ.
Σ_{k=0}^{n} C(n, k) = 2ⁿ.
Inclusion–exclusion: |A ∪ B ∪ C| = |A| + |B| + |C| − |A ∩ B| − |A ∩ C| − |B ∩ C| + |A ∩ B ∩ C|.

Mawazo potofu ya kawaida

Using permutations when order does not matter.
Thinking a pigeonhole argument must say which box contains two objects.

Inaelekea kwa

Conditional probability and independence Recurrences and generating functions Graph theory

Ugunduzi wa hiari

Enumerative combinatorics Generating functions turn counting problems into algebra.
← Systematic listing and counting

Njia za kujifunza

Usomaji zaidi

Arthur Engel, Problem-Solving Strategies (1998)