How many natural numbers less than 107
exist such that they digits are pair-different?
Using induction prove the binominal Newton's
formula.
Prove that for each natural number n, holds
(n over 0) 2 + (n over 1) 2 + ... + (n over n)
2 = (2n over n)
We say that two k-element variations over the
set
{1, 2, ..., n} are equivalent if they are different in terms of
elements' order. How many abstraction classes does this equivalence
relation have. Justifyyour answer.
In how many ways can a natural number n
be presented as a sum of two other natural numbers?
Using inequality n! > (n/e)n,
prove that for any natural number k < n, holds (n/k)k ≤
(n over k) ≤ (en/k)k .
Prove that in every 1000 people's group there
are at
least three people who celebrate birthday on the same day.
In how many ways can we choose 3 different
numbers from the set {1, 2, ..., 100} such that their arithmetic sum is
even?
Prove that for every natural number n, the
following holds:
(n over 1) + 2 (n over 2) + 3 (n over 3) +
... + n (n over n) = n 2(n-1),
(n over 1) - 2 (n over 2) + 3( n over 3) +
... +(-1) (i-1) i(n over i) + ...
+(-1) (n-1) n(n over n) = 0
2 ⋅
1 ⋅ (n over 2) + 3 ⋅
2 ⋅ (n over 3) + ... + n (n-1) (n over n) =
n(n-1) 2(n-2) .