Daily math problems for July 30, 2026 — Algebra, Calculus, Combinatorics & more

$\mathrm{deg}=1+2=3$.

$f$ is undefined at $x=2$, where the denominator vanishes.

The coefficient of $x^2$ corresponds to $k=2$: $\left(\genfrac{}{}{0px}{}{4}{2}\right)=6$.

$\bullet$ Simplify each entry. Row 1: $\cos 2024\pi =1$, $\tan \pi =0$, $i^{2024}=1$. Row 2: $\sin 0=0$, ${\log }_{2}16=4$, $0!=1$. Row 3: $\ln 1=0$, $1$, $\cos 360\mathrm{°}=1$.

$\bullet$ The matrix becomes:

$$\left(\begin{array}{ccc}1& 0& 1\\ 0& 4& 1\\ 0& 1& 1\end{array}\right)$$

$\bullet$ Two zeros in column 1, so expand along it:

$$\det =1\cdot \mid \begin{array}{cc}4& 1\\ 1& 1\end{array}\mid =4-1=3$$

$f^{\mathrm{\prime }}(x)=2x\ln x+x^2\cdot \frac{1}{x}=2x\ln x+x=x(2\ln x+1)$.

$\bullet$ Decode the counts:

$${\log }_{3}9=2,i^{100}+3=1+3=4,2!\cos 0=2$$

$\bullet$ So $(a,b,c)=(2,4,2)$, total $8$ balls. Count favorable pairs:

$$\left(\genfrac{}{}{0px}{}{2}{2}\right)+\left(\genfrac{}{}{0px}{}{4}{2}\right)+\left(\genfrac{}{}{0px}{}{2}{2}\right)=1+6+1=8$$

$\bullet$ Total number of pairs:

$$\left(\genfrac{}{}{0px}{}{8}{2}\right)=28$$

$\bullet$ Probability:

$$P={\displaystyle \frac{8}{28}}={\displaystyle \frac{2}{7}}$$

LHS: $2\cdot 1+1=3$. RHS: $1+a$. Setting equal: $a=2$.

$!4=4!(1-1+{\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{6}}+{\displaystyle \frac{1}{24}})=24\cdot {\displaystyle \frac{9}{24}}=9$.

The units are the residues coprime to $8$: $\{1,3,5,7\}$, four elements.

$a=5$, $b=3$, $c=\sqrt{25-9}=4$. So $e={\displaystyle \frac{4}{5}}$.