Coefficients binomiaux

n, p ∈N∗
n
X
i=0
p
Y
j=1
(i+j)
n∈N∗
n
X
k=1
(−1)k+1
kn k=
n
X
k=1
1
k
n∈Nn≥2
n
∀k∈ {2, . . . , n −1}, n n
k
n
∃k∈ {2, . . . , n −1}, n n
k
n∈N∗
∀1≤k≤n, n
k=n−k+ 1 kn k−1
k1≤k≤n/2
n
k−1<n k
k n/2≤k≤n−1
n
k+ 1<n k
∀n∈N∗,2n
n≥22n 2n+ 1

n
X
i=0
p
Y
j=1
(i+j)
=
n
X
i=0
(i+p)!
i!
n
X
i=0
p
Y
j=1
(i+j)
=p!
n
X
i=0 i+p
i
n
X
i=0
p
Y
j=1
(i+j)
=p!p+n+ 1
n=(p+n+ 1)! (p+ 1)n!
n≥1
n+1
X
k=1
(−1)k+1
kn+ 1 k=
n+1
X
k=1
(−1)k+1
kn k+
n+1
X
k=1
(−1)k+1
kn k−1
n+1
X
k=1
(−1)k+1
kn k−1=
n+1
X
k=1
(−1)k+1
n+ 1 n+ 1 k=1 n+ 1 −(1 −1)n+1 n+ 1 =1 n+ 1
n+1
X
k=1
(−1)k+1
kn+ 1 k=
n+1
X
k=1
1
k
n
n
k=n kn−1 k−1
kn
k=nn−1 k−1
n kn
kn
k k < n
nn
k
n p
n p < n n n
p
m=1
nn p
(n−1)! = m.p!(n−p)!
p n n =pq q
(pq −1)! = mp! (p(q−1))!
(pq −1)! (p(q−1))!
p p, 2p, . . . (q−1)p
(pq −1)! = ka (p(q−1))! = kb
k p a b
p
a=mp!b
a p
n!
k!(n−k)! =(n−k+ 1) k
n!
(k−1)!(n−k+ 1)!
1≤k≤n/2 2k < n + 1 n−k+ 1 > k
n
k=n−k+ 1 kn k−1>n k−1
n
k=n n−k

n
2n
X
k=0 2n
k= (1 + 1)2n= 22n
n
∀0≤k≤2n, 2n
k≤2n n
22n≤(2n+ 1)2n
n