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对于组合数恒等式的证明无固定的方法, 使得人们常感到无从下手.下面介绍构造概率 模型证明组合恒等式几例,供读者参考. 例1 求证:Cn0+Cn1+Cn2+…+Cnn=2n. 证明 设事件A在一次试验中发生的概率 为1/2,那么n次独立重复试验中恰好发生k次的 概率是:PA(k)=Cnk(1/2)k·(1-1/2)n-k=1/2nCnk. 令k=0,1,2,…,n,并求和得 即 Cn0+Cn1+Cn2+…+Cnn=2n. 例2 求证:(Cn0)2+(Cn1)2+…+(Cnn)2= C2nn. 证明 设一个口袋中有n个白球n个红 球,任取n个球,求A={至少有一个白球}的概
There is no fixed method for the proof of combinatorial number identity, which makes people often feel that there is no way to start. Here are some examples of constructing probabilistic models to prove combinatorial identities for readers’ reference. Example 1 Verification: Cn0+Cn1+Cn2+...+Cnn=2n. The probability of occurrence of event A in a test is 1/2, so the probability of exactly k occurrences in n independent repeated tests is: PA(k)=Cnk(1/2)k(1-1/2)nk =1/2nCnk. Let k=0,1,2,...,n, and add it to be Cn0+Cn1+Cn2+...+Cnn=2n. Example 2 Verification: (Cn0)2+(Cn1)2+...+ (Cnn)2= C2nn. Prove that there are n white balls n red balls in a pocket, take n balls, ask for A={at least one white ball}