From the set of numbers 1 to \(2n\), \(n + 1\) numbers are chosen. Prove that among the chosen numbers there are two, one of which is divisible by another.
How many distinct seven-digit numbers exist? It is assumed that the first digit cannot be zero.
We call a natural number “fancy”, if it is made up only of odd digits. How many four-digit “fancy” numbers are there?
We are given 51 two-digit numbers – we will count one-digit numbers as two-digit numbers with a leading 0. Prove that it is possible to choose 6 of these so that no two of them have the same digit in the same column.
Prove that amongst any 11 different decimal fractions of infinite length, there will be two whose digits in the same column – 10ths, 100s, 1000s, etc – coincide (are the same) an infinite number of times.
The number of permutations of a set of \(n\) elements is denoted by \(P_n\).
Prove the equality \(P_n = n!\).
How many six-digit numbers exist, for which each succeeding number is smaller than the previous one?
Why are the equalities \(11^2 = 121\) and \(11^3 = 1331\) similar to the lines of Pascal’s triangle? What is \(11^4\) equal to?
How many four-digit numbers can be made using the numbers 1, 2, 3, 4 and 5, if:
a) no digit is repeated more than once;
b) the repetition of digits is allowed;
c) the numbers should be odd and there should not be any repetition of digits?
Write in terms of prime factors the numbers 111, 1111, 11111, 111111, 1111111.