Yahoo Answers: Answers and Comments for Prove the identity Sigma i=0..n C(i+k1,k1) = C(n+k,k). C(n,r) denotes n choose r.? [Mathematics]
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From Anonymous
enCA
Thu, 03 Apr 2008 17:13:41 +0000
3
Yahoo Answers: Answers and Comments for Prove the identity Sigma i=0..n C(i+k1,k1) = C(n+k,k). C(n,r) denotes n choose r.? [Mathematics]
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https://ca.answers.yahoo.com/question/index?qid=20080403171341AAd3MMt
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From Anonymous: By induction
 first take n = 0
then we have...
https://ca.answers.yahoo.com/question/index?qid=20080403171341AAd3MMt
https://ca.answers.yahoo.com/question/index?qid=20080403171341AAd3MMt
Fri, 04 Apr 2008 06:42:28 +0000
By induction
 first take n = 0
then we have
sum (i = 0) = C(k1,k1) = 1 = C(k,k)
so the result is correct for n = 0
 Suppose the result is correct for n = 0,1,...,m
we have to prove the result is correct for n = m+1
now
sum(i = 0,1,...,m+1) C(i+k1,k1)
= sum (i =0,1,...,m) C(i+k1,k1) + C(m+1+k1,k1)
= (induction for the first part)
= C(m+k,k) + C(m+k,k1)
= formulas
= (m+k)!/[k!m!] + (m+k)!/[(k1)!(m+1)!]
= [(m+k)!*(m+1) + (m+k)!k]/[k!(m+1)!]
=(m+1+k)!/[k!(m+1)!]
= C(m+1+k,k)
so the formula also holds for n = m+1
QED