Proving a Summation Identity

Please see the attached file for the fully formatted problem.

Proving a Summation Identity.

Please see the attached file for the fully formatted problem.

Estimation : Binomial Distribution

Suppose T is a random variable such that P(T=k) = (k-1)C(r-1) * p^r * (1-p)^(k-r) (It is a negative binomial distribution.). I am trying to find the expected value E(r/T) (which is equal to r * E(1/T)) By (k-1)C(r-1) I mean (k-1)!/[(r-1)!*(k-r)!].

n-bit binary strings with fixed digits

I need to show that the number of n-bit binary strings that contain exactly two occurrences of 10 is C(n+1, 5)

This is a problem in counting. It is a beginner problem in probability theory

How many 5 card poker hands are there? Note a deck of 52 cards is assumed.