Problems

Age
Difficulty
Found: 1947

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?

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?

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) \(\binom{n}{0} + \binom{n}{1} + \dots + \binom{n}{n}\).