<< set_theory

# Total Permutation

• Case of all infinite:

  Total permutaton of infinite of

• Case of all finite:

  For multiple set A = \({n_1\cdot a_1, n_2\cdot a_2, n_3\cdot a_3, ..., n_k\cdot a_k}\), where \(\sum n_i = n\), the total permutation of set A:

  \[ \cfrac{n!}{n_1! n_2! ... n_k!} \]

  ==todo:finish this formula== ==todo:find permutation symbol in latex==

  This is the same with the following case: dispense n different objects to k different blocks.==todo:why?==

• Case of both finte and infinte

# Combination

• Case of all inifnite: \[ {n-1+r\choose r} \]

==todo:add explanation using the element-bar persepective== ==todo:add some examples to explain this, and explain how the traditional perspective will explain these problems?==

Ex. How many times the inner for loop will execute(the value of k)?

for(k=0, i1=0; i1<=n; i1++)
    for(i2=0; i[2]<=i[1]; i[2]++)
        ...
            for(im=0; i[m]<=i[m-1]; i[m]++)
                k++

\[ {n+m\choose m} \]

2. inclusion-exclusion principle

==todo:include t==