Key takeaway: Pascal’s triangle is a triangular array where each entry is the sum of the two entries above it, and row n lists the binomial coefficients C(n, 0) through C(n, n).
Grade 12 – PrecalculusA-APR.C.5Expressions and Equations
Definition of Pascal's Triangle
Pascal’s triangle starts with a single 1 at the top. Every new row begins and ends with 1, and each number inside is the sum of the two numbers diagonally above it. Rows are counted from row 0 at the top.
Beneath this simple rule hides a lot of mathematics. Row \(n\) holds the coefficients of \((a+b)^n\), the numbers of ways to choose \(k\) items from \(n\), and sums that double from one row to the next. Diagonals contain the counting numbers, the triangular numbers and more. The array was known in several cultures long before it carried Blaise Pascal’s name.
Formula and notation
The entry in row \(n\), position \(k\) (both counted from 0) is
\[\binom\,{n}{k}=\binom\,{n-1}{k-1}+\binom\,{n-1}{k},\qquad \sum_{k=0}^{n}\binom\,{n}{k}=2^{n}.\]
The first formula is the “add the two above” rule; the second says that row \(n\) sums to \(2^n\).
Key facts
- Each row is a mirror image of itself, because C(n, k) = C(n, n − k).
- Row 5 is 1, 5, 10, 10, 5, 1, and its entries add to 32.
- Row n has n + 1 entries, numbered from position 0 to position n.
- Reading row n gives the coefficients for expanding a binomial to the power n.
Pascal's Triangle example
Build row 7 of the triangle from row 6, then find the sum of row 7 and the value of \(\binom\,{7}{3}\).
-
Row 6 is 1, 6, 15, 20, 15, 6, 1.
-
Add neighbors: \(6+1=7\), \(15+6=21\), \(20+15=35\), and by symmetry \(35, 21, 7\). Row 7: 1, 7, 21, 35, 35, 21, 7, 1.
-
Row sum: \(2^7=128\); check \(1+7+21+35+35+21+7+1=128\).
-
Position 3 of row 7 is 35.
Row 7 adds to 128, and \(\binom\,{7}{3}=\)35.
Common mistake
Watch out
Cyrus wants \(\binom\,{7}{3}\) and reads the third number of row 7, which is 21.
Why it is wrong
Positions are counted from 0, and rows too. The entry \(\binom\,{7}{3}\) is the fourth number: 1, 7, 21, 35. Counting from 1 is the usual slip in Pascal’s triangle.
Related terms and lessons
Browse the glossary: A–Z list · Precalculus vocabulary · Expressions and Equations