Discrete-math 9

Combinatorics

\begin{matrix} Answer the question: how many ways to do smth there is? \end{matrix}

\begin{matrix} How many four digit numbers there are, comprised of digits 1, 2, 3, 4, 5 ? \\ Solution: \\ 5^{4} \\ How many such numbers with at least one 3 are there? \\ Let us count the number of numbers without 3 \\ 4^{4} \\ ⟹ Number of numbers with at least one three is 5^{4} - 4^{4} \end{matrix}

\begin{matrix} How many ways to place 2 black rooks on a chess board \\ such that they don’t threaten each other? \\ Solution: \\ \frac{64 * (64 - 15)}{2!} = \frac{64 * 49}{2} = 32 * 49 \end{matrix}