• Properties of Binomial Coefficient

    \(1. \;  \binom{n}{k}\binom{k}{m} = \binom{n}{m}\binom{n-m}{k-m}\)


    Combinatorial proof:

    It's election time! There are k city councillors that need to be chosen, but there are n condidates. As well, Mayor needs m councillors to be part of his executive committee.

  • Properties of Binomial Coefficient

    \(2. \; \binom{n}{k} = \frac{n-k+1}{k}\binom{n}{k-1}\)


    Combinatorial proof:

    A king ordered his advisors to select a new panel of k advisors from a group of n candidates. However, he decided to first see a list of all possible panels.

  • Combination and Permutations

    \(1. \; \binom{2n}{2} = 2\binom{n}{2} + n^{2}\)


    Proof 1:

    LHS: chooses 2 people from 2n people.

    RHS: We first decides whether to choose both people from the first or second half. If so, we have two halves of n people.  We could pick either the first half or the second half. Then we choose 2 from that n people. Also, we could choose one from the first n, and one from the second, giving n * n = n2 ways.

    As we have counted the same things on both sides, therefore, LHS equals RHS, and so

  • Expected Value

    1. Show E(N)=6 , where  N is the number of tries until getting the first 3 using a fair dice

    For a fair dice , the chances of getting a 3 are:

    1/6, on the 1st try

    1/6 * (5/6), on the 2nd try as the probability of failure on the 1st try is 5/6