MTH 225- Final Exam Guide - Comprehensive Notes for the exam ( 32 pages long!)
Document Summary
Video lecture: definition: a combinatorial identity is an equation that often times involves binomial, combinatorial identities using algebraic proof: this is using the recall statements and, 1. Section 1. 4- combinatorial proofs showing that the left hand side equals the right hand side: recall: (cid:4672)(cid:4673)= ! (cid:4666) (cid:4667)!! and 0! = 1 coefficients. (cid:4672)(cid:4673)= (cid:4672) (cid:4673: example: (cid:4672)(cid:882)(cid:4673)=(cid:883, proof: (cid:4672)(cid:882)(cid:4673)= ! (cid:4666) (cid:2868)(cid:4667)! (cid:2868)! (cid:2868)! (cid:2869)(cid:2869) (cid:2869, (cid:4672)(cid:4673)=(cid:883, example: (cid:4672)(cid:883)(cid:4673), proof: (cid:4672)(cid:883)(cid:4673)= ! (cid:4666) (cid:2869)(cid:4667)! (cid:2869), (cid:4666) (cid:2869)(cid:4667)! (cid:4666) (cid:2869)(cid:4667), example: (cid:4672)(cid:4673)=(cid:4672) (cid:883) You count it in 2 different ways; first to equal left hand side, then again to equal right. Since you count the same thing, then they are equal: simple question: how many dots are here. Second says 3 rows of 4 (which is 12). N} to not be in our set. This leaves exactly k elements for our set. If they were sets, they would all be equivalent.
