Problems

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?

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?

Find $m$ and $n$ knowing the relation $(\genfrac{}{}{0ex}{}{n+1}{m+1}):(\genfrac{}{}{0ex}{}{n+1}{m}):(\genfrac{}{}{0ex}{}{n+1}{m-1})=5:5:3$.

Write at random a two-digit number. What is the probability that the sum of the digits of this number is 5?

A player in the card game Preferans has 4 trumps, and the other 4 are in the hands of his two opponents. What is the probability that the trump cards are distributed a) $2:2$; b) $3:1$; c) $4:0$?

The numbers $1,2,\dots ,99$ are written on 99 cards. Then the cards are shuffled and placed with the number facing down. On the blank side of the cards, the numbers $1,2,\dots ,99$ are once again written.

The sum of the two numbers on each card are calculated, and the product of these 99 summations is worked out. Prove that the end result will be an even number.

Prove that any $n$ numbers $x_1,\dots ,x_n$ that are not pairwise congruent modulo $n$, represent a complete system of residues, modulo $n$.

Prove that for any natural number there is a multiple of it, the decimal notation of which consists of only 0 and 1.

Without calculating the answer to $2^{30}$, prove that it contains at least two identical digits.

Let the number $\alpha$ be given by the decimal:

a) $0.101001000100001000001\dots$;

b) $0.123456789101112131415\dots$.

Will this number be rational?