1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Simplifying a Product

  1. Jun 22, 2016 #1
    1. The problem statement, all variables and given/known data
    This is a child thread I'm creating from a previous topic:

    In that thread, I was helped to come up with the expression for the number of arrangements of R distinct types of objects given the number of objects for each object type: {r_i} I'm just trying to simplify it now. I wanted to double check the work. Something seems off.

    2. Relevant equations
    $$\prod {_{n - \sum{r_{i-1}}}C_{r_j}}$$
    or more specifically,
    $$\prod_{j=1}^{R} {_{n - \sum_{i=1}^{j}{r_{i-1}}}C_{r_j}}$$
    $$r_0 = 0$$
    3. The attempt at a solution

    $$_{m}C_{k} = \frac{m!}{k!(m-k)!}$$
    $$\therefore \space\space\space\space\prod_{j=1}^{R} \frac{(n - \sum_{i=1}^{j}{r_{i-1}})!}{r_j!(n - \sum_{i=1}^{j}{r_{i-1}} - r_j)!}$$

    $$\frac{(n - 0)! \cdot (n - r_1)! \cdot (n - r_1 - r_ 2)! ... }{(r_1! \cdot r_2! \cdot r_3! ... ) [(n - r_1)! \cdot (n - r_1 - r_2)! \cdot (n - r_1 - r_2 - r_3)! ... ] }$$

    $$n!\prod_{j=1}^{R} \frac{1}{r_j!}$$

    Let me know if I can clear anything up!
  2. jcsd
  3. Jun 23, 2016 #2


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Provided it is the case that ##\sum_{i=1}^Rr_i=n##, that looks correct to me.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted