## Combinatorics (VI)

29 Dec, 2022 at 15:21 | Posted in Statistics & Econometrics | 4 Comments

In my library, there are n philosophy books and six economics books. If yours truly can choose two books of each type in 150 ways, how many philosophy books are there in my library?

1. I am still thinking in terms similar to diagram structures in combinatorics. Not as elegant as your calculation. (:-)

• how many history books? The business history review, 1926-present is a first rate sourrce that we ignore in the rwer world.

2. C(n,r) = n!/(r!(n-r)!)

There are 15 combinations of 6 econ books taken 2 at a time.
C(6,2) = 6!/(2!4!) = 30/2 =15

That means that if x is the number of phil books, C(x,2) =10 (150/15)
C(5,2) = 5!/(2!3!) = 20/2 = 10
There are 5 philosophy books.

— John Lounsbury

• Right answer! My calculation, however, goes like this:
nC2*6C2 =150 =>
n!/(n-2)!*2 * 6!/(6-2)!*2! = n(n-1)/2! * 15 =>
n(n-1)/2 =150/15 => n(n-1) = 20 => n^2 – n -20 = 0 => (n-5)(n+4) = 0  =>  n = 5.

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