Category Archives: Combinatorics

Nine light bulbs are equally spaced around a circle. When the power is turned on, each of the nine light bulbs turns blue or red, where the color of each bulb is determined randomly and independently of the colors of … Continue reading →

Let $f(n)$ be the number of binary strings with length as $n$ that they do not contain a sub-string as “$010$” nor “$101$”. Prove that for $n\ge 3$: $f(n)=f(n-1)+f(n-2)$ 🔑 Proof: For $n=1$, we have $2$ possible binary strings, as … Continue reading →

Find the number of ways to write $24$ as the sum of at least three positive integer multiples of $3$. For example, count $3+18+3$, $18+3+3$, and $3+6+3+9+3$, but not $18+6$ or $24$. (PurpleComet 2023, Middle School, Problem 8) Click here … Continue reading →

A soccer team has $8$ players. They need to form a starting lineup with one goalkeeper, one captain, one vice-captain, three unique field players, and two bench players. However, two specific players do not want to be captain, vice-captain, or … Continue reading →