Daily math problems for May 5, 2026 — Combinatorics, Compound Interest, Exponential Functions

Alice must take the window seat (1 way). Bob must take the aisle seat (1 way). Carol takes the middle (1 way). Only $1\times 1\times 1=1$ ordering satisfies both preferences.

More generally — when each person's preference is for a distinct seat, only the permutation that matches preferences exactly is 'fully happy'.

We need $t$ such that $(1.07{)}^t=2$, i.e. $t={\displaystyle \frac{\ln 2}{\ln 1.07}}\approx {\displaystyle \frac{0.693}{0.0677}}\approx 10.24$ years.

The banker's shortcut: the Rule of 72 — divide 72 by the interest rate. $72\mathrm{/}7\approx 10.3$. Quick, accurate enough.

Take $\ln$ of both sides: $(x+1)\ln 2=x\ln 3$.

Expand: $x\ln 2+\ln 2=x\ln 3$.

Collect $x$: $\ln 2=x(\ln 3-\ln 2)$.

So $x={\displaystyle \frac{\ln 2}{\ln 3-\ln 2}}\approx {\displaystyle \frac{0.693}{0.405}}\approx 1.71$.