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Multiply by 6 for the 6 possible blocks:
This fraction is already in simplest form.
$$Question: An angel investor is considering funding one of 5 early-stage tech startups, each focusing on a unique field: AI, biotech, clean energy, quantum computing, and blockchain. If the investor randomly selects 3 startups to fund, what is the probability that both the AI and biotech startups are among the selected ones?
To solve this, we begin by computing the total number of ways to choose 3 startups from 5:
Next, we count the number of favorable outcomes: those that include both the AI and biotech startups. If AI and biotech are already selected, the third startup must be one of the remaining 3: clean energy, quantum computing, or blockchain. So, there are:
favorable outcomes. Therefore, the desired probability is:
Question: A UX designer is creating a mobile app with 4 home screen widgets: a calendar, a to-do list, a weather widget, and a fitness tracker. If the designer randomly arranges these widgets in a 2x2 grid, what is the probability that the calendar and weather widget are adjacent (share a side)?
We begin by computing the total number of ways to arrange 4 distinct widgets in a 2x2 grid:
Now we count the number of favorable arrangements where the calendar and weather widget are adjacent. We treat the calendar and weather widget as a single block and count the number of ways to place this block and the other two widgets in the grid.
There are 4 positions in the 2x2 grid. For each position of the calendar-weather block, we check which adjacent positions are available: