There are 17 carriages in a passenger train. How many ways can you arrange 17 conductors around the carriages if one conductor has to be in each carriage?
The number of permutations of a set of \(n\) elements is denoted by \(P_n\).
Prove the equality \(P_n = n!\).
How many ways can you choose four people for four different positions, if there are nine candidates for these positions?
There are \(n\) points on the plane. How many lines are there with endpoints at these points?
On a plane \(n\) randomly placed lines are given. What is the number of triangles formed by them?
Prove the validity of the following formula of Newton’s binom \[(x+y)^n = \binom{n}{0}x^n + \binom{n}{1}x^{n-1}y + \dots + \binom{n}{n}y^n.\]
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?
Think of a way to finish constructing Pascal’s triangle upward.
Calculate the following sums:
a) \(\binom{5}{0} + 2\binom{5}{1} + 2^2\binom{5}{2} + \dots +2^5\binom{5}{5}\);
b) \(\binom{n}{0} - \binom{n}{1} + \dots + (-1)^n\binom{n}{n}\);
c) \(\binom{n}{0} + \binom{n}{1} + \dots + \binom{n}{n}\).