कक्षा 9–12Discrete Mathematicsनियोजित
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
इस विषय का पाठ अभी लिखा नहीं गया है। नीचे दी गई रूपरेखा पाठ्यक्रम-मानचित्र का हिस्सा है और इसकी समीक्षा नहीं हुई है।
इस पर आधारित
15 अनिवार्य पूर्वापेक्षाएँ (बिल्कुल नीचे तक)
आप सीखेंगे
- 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.
परिभाषा
- 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.
प्रमेय
- 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|.
आम भ्रांतियाँ
- Using permutations when order does not matter.
- Thinking a pigeonhole argument must say which box contains two objects.
आगे ले जाता है
वैकल्पिक खोजें
- Enumerative combinatorics Generating functions turn counting problems into algebra.
- ← Systematic listing and counting
अधिगम पथ
आगे पढ़ने के लिए
- Arthur Engel, Problem-Solving Strategies (1998)