Page 154 - Fixtures

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.

C(r + k,k) = C(r + k,r + k − k) = C(r + k,r)

. n C (r + k,k) = n + k,r)

∑ ∑ C (r
k=0 k=0

. n + k,r) = C (r + n + 1, r )

∑C (r
k=0

.C(r + n + 1,r) = C(r + n + 1,n)

. n C (r + k,k) = C (r + n + 1, n )

∑
k=0

.

[Vandermond`s Identity]

r ≤m+n r ,n ,m

r

C(m + n,r) = ∑C (m,r − k)C (n,k)
k =0

A =m BA :
A∪B
r, .B =n
:
.C(m + n,r)
A∪B
A∪B r ,

.0 ≤ k ≤ r A r −k B k

r , .C(n,k)C(m,r − k)