Question 628135: I have some equations in the following format (finding the nth term of an algebraic expression)
( n ) a^(n-(r-1)) * b^(r-1)
r-1
I don't understand what the (n & r-1) terms in the parenthesis at the beginning of the expression are. From the solution, it suggests that the first term ends up resolving to a division of factorials such as:
(6! / ((6-3)! * 3!)) where n=6 and r=4...
I just dont understand how they got from the first form to the factorial form.
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
I think what you are looking at is  (read: "n choose r - 1") which is the number of combinations of  things taken  at a time. Such as, how many ways can I choose from 10 different books on a shelf if I take them 3 at a time and I don't care what order the three are in when I get them in my hand?
In general,  is calculated by !) . Hence, your !}\ =\ 20) is exactly correct given  and  .
Here's the logic: Let's say you have 6 things and want to choose 3 of them. There are 6 ways to choose the first one, then since you didn't replace the first one you chose, there are 5 ways to choose the second one for each one of the 6 ways to choose the first one. Then there are 4 ways to choose the third one for each of the 30 ways to pick the first two, which works out to  which comes from !}\ =\ \frac{6\ \times\ 5\ \times\ 4\ \times\ 3\ \times\ 2}{3\ \times\ 2}) . But that number is too large by a factor of the number of ways to arrange the three things in your hand, namely  , and that is where the other denominator factor comes from.
You might want to compare this to the number of permutations of things taken  at a time. With permutations, order matters. Such as you have 20 people in your club and you want to know how many different ways you can select a President, Secretary, and Treasurer. Here order matters because Suzy being the president is a different outcome than Suzy being the Secretary, for example. Permutations are calculated !}) . See the difference?
By the way, this is also the th coefficient (counting from zero) of the binomial expansion of ^n) and the th element of the th row of Pascal's Triangle.
John
My calculator said it, I believe it, that settles it
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