Monday, April 16, 2018

Combinatorics Derivations


Combinatorics Derivations



The definition of the combination function is:



C(n, r) = n! / (r! *  (n – r)!)



Today I am going to mathematically verify three equivalents in combinatorics.



Newton’s Identity



C(n,r) * C(r,k) = C(n,k) * C(n – k, r – k)



Here I am going to assume that n > r > k.



Hence:



C(n, r) * C(r, k)

= n! / (r! * (n – r)!) * r! / (k! * (r – k)!)

= n! / (n – r)! * 1 /  (k! * (r – k)!)



Rearrange:



= n! / k! * 1 / ((n – r)! * (r – k)!)



Multiply by (n – k)!/(n- k)!:



= n! / (k! * (n – k)!) * (n – k)! / ((n – r)! * (r – k)!)



Observe that (n – k) – (r – k) = n – k – r + k = n – r.  Hence,



= C(n,k) * C(n – k, r – k)   QED



Pascal’s Identity



C(n,r) = C(n-1, r) + C(n – 1, r – 1)



I’m going to start with C(n-1, r) + C(n – 1, r – 1)



C(n-1, r) + C(n – 1, r – 1)



= (n – 1)! / (r! * (n – 1 – r)!) + (n – 1)! / ((r – 1)! * (n – r)!)



= (n – 1)! / (r! * (n – 1- r)!) + (r * (n – 1)!) / (r! *(n – r)!)



= ((n – 1)! * (n – r)) / (r! * (n – r)!) + (r * (n – 1)!) / (r! * (n – r)!)



= (n * (n – 1)! – r * (n – 1)! + r * (n – 1)!) / (r! * (n – r)!))



= (n * (n  - 1)!) / (r! * (n – r)!)



= n! / (r! * (n – r)!)



= C(n, r)  QED



Combinatorial Proof



C(m + n, 2) – C(m, 2) – C(n, 2) = m * n



C(m + n, 2) – C(m, 2) – C(n, 2)



Note that 2! = 2



= (m + n)! / (2 * (m + n – 2)!) – m! / (2 * (m – 2)!) – n! / (2 * (n – 2)!)



= ( (m + n)! * (m – 2)! * (n – 2)! – m! * (m + n – 2)! * (n – 2)!  - n! * (m – 2)! * (m + n – 2)!) / (2 * (m + n – 2)! * (m – 2)! * (n – 2)!)



= ( (m + n) * (m + n – 1) *(m + n – 2)! * (m – 2)! * (n – 2)! – m * (m – 1) *(m – 2)! * (n – 2)! * (m + n – 2)! – n * (n – 1) * (n – 2)! * (m – 2)! * (m + n – 2)! )

/  (2 * (m + n – 2)! * (m – 2)! * (n – 2)!)



= ( (m + n)*(m + n – 1) – m * (m – 1) – n * (n – 1) ) / 2



= ( m^2 + m*n – m + m*n + n^2 – n – m^2 + m – n^2 + n) / 2



= (2 * m * n) / 2



= m * n   QED



Eddie



Source where I got the identities from:

V.K. Balakrishnan Schaum’s Theory and Problems: Combinatorics including concepts of Graph Theory  McGraw-Hill, Inc. New York: 1995  ISBN 0-07-003575-X



The derivation and proof details are my work.  



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