Problems

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?

Out of two mathematicians and ten economists, it is necessary to form a committee made up of eight people. In how many ways can a committee be formed if it has to include at least one mathematician?

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?

On two parallel lines $a$ and $b$, the points $A_1,A_2,\dots ,A_m$ and $B_1,B_2,\dots ,B_n$ are chosen, respectively, and all of the segments of the form $A_i{B}_j$, where $1\le i\le m$, $1\le j\le n$. How many intersection points will there be if it is known that no three of these segments intersect at one point?

Prove the validity of the following formula of Newton’s binom $$(x+y{)}^n=(\genfrac{}{}{0ex}{}{n}{0})x^n+(\genfrac{}{}{0ex}{}{n}{1})x^{n-1}y+\cdots +(\genfrac{}{}{0ex}{}{n}{n})y^n.$$

How many rational terms are contained in the expansion of

a) $(\sqrt{2}+\sqrt[4]{3}{)}^{100}$;

b) $(\sqrt{2}+\sqrt[3]{3}{)}^{300}$?

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?