Pascal's Triangle

Combinatorics · Axiom Academy

Discover the hidden patterns in one of mathematics' most elegant structures. Let's construct Pascal's Triangle row by row. Use the control to add rows and watch how the triangle grows. The key to Pascal's Triangle: each interior number is the sum of the two numbers directly above it. Click on any cell to see how it's formed! Now it's your turn! Drag the correct numbers to complete row 4 of Pascal's Triangle using the addition rule. Pascal's Triangle is full of surprising patterns. Select a pattern below to see it highlighted in the triangle! Perfect mirror across the center Pascal's Triangle isn't just pretty—it's deeply connected to binomial coefficients and combinatorics! Each number in Pascal's Triangle tells you "how many ways" to choose things. The number in row n , position k equals , which means "how many ways can you choose k items from n items?" Row 4, position 2 = 6 This means there are 6 ways to choose 2 items from 4 items. If you have 4 friends and want to invite 2 to dinner, you have 6 different pairs to choose from! When you expand , the coefficients are exactly the numbers in row n of Pascal's Triangle. For example: Pascal's Triangle connects counting, algebra, probability, and geometry. It appears in probability theory, computer science, fractal geometry, and even the stock market! Understanding this triangle unlocks a deeper appreciation of mathematical patterns.

This is the written version of the interactive lesson above. See the full Combinatorics course.