six students, two cannot be together, four double rms, two single rms, how many ways

Six students need to be placed in a dormitory. There are four double rooms, two single rooms, and two students cannot be placed together, how many ways are there to place the students?

Solving an equation using logarithms.

Solve for x: 1.8^(2x+3) = 0.72^1/3

Combinations: Forming groups from a bigger group

If you have the numbers 1-20,how many groups of five numbers can be formed? And please list the groups.

Proving a Summation using an Identity

Please see the attached file for the fully formatted problems. I’m trying to prove the following: using the following identity:

One equality

Hello! The attached file is the answer an OTA gave me to a problem I was having. There is one line in the reasoning I do not understand, and I have highlighted it in red. I would be very grateful if someone could explain it to me.

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.