How to Solve Pascal’s Triangle?

Pascal’s triangle can be used for many purposes in mathematics. It is used in probability, to find the number of combinations, in the binomial expansion of a polynomial, and can be used to find the Fibonacci series.

Related Topics

• The Binomial Theorem

A step-by-step guide to Pascal’s triangle

Pascal’s triangle is an arrangement of numbers in a triangular array such that the numbers at the end of each row are \(1\) and the remaining numbers are the sum of the two nearest numbers in the row above. The number of elements in the \(nth\) row is equal to \((n + 1)\) elements.

Pascal’s triangle can be easily constructed by adding the pairs of consecutive numbers in the previous lines and writing them in the new line.

we use Pascal’s triangle formula to fill the number in the \(nth\) column and \(mth\) row of Pascal’s triangle. The formula requires the knowledge of the elements in the \(\left(n-1\right)^{th}\) row, and \(\left(m-1\right)^{th}\) and nth columns. The elements of the \(n^{th}\) row of Pascal’s triangle are given by, \(^nC_0\), \(^nC_1\), \(^nC_2\), …, \(^nC_n\). The formula for Pascal’s triangle is:

\(\color{blue}{^nC_m\:=\:^{n-1}C_{m-1}+\:^{n-1}C_{m}}\)

where

•  \(^nC_m\) showes the \(\left(m+1\right)^{th}\) element in the \(n^{th}\) row.
• \(n\) is a non-negative integer
• \(0≤ m≤n\).

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Pascal’s triangle pattern

Pascal’s triangle has various patterns within the triangle that were discovered and explained by Pascal himself or known before him. Some of Pascal’s triangle patterns are:

• The sum of values in the \(n^{th}\) row is \(2^n\).
• If a row has a second element that is a prime, then all the following elements in the row are divisible by that prime (not including \(1s\)). ex. \(1 5 10 10 5 1\).
• By adding the diagonal elements of Pascal’s triangle, we get the Fibonacci series.

Pascal’s Triangle – Example 1:

Find the \(3rd\) element in the \(4th\) row.

Solution:

This means you want to calculate \(^4C_2\:\). Then according to the formula:

\(^4C_2\:=^{4-1}C_{2-1}+^{4-1}C_2\:\:\)

\(^4C_2=^3C_1+^3C_2\)

So, this means we need to add the \(2nd\) element in the \(3rd\) row (i.e. \(3\)) with the \(3rd\) element in the \(3rd\) row (i.e. \(3\).). So that would be our answer:

\(^4C_2=3+3=6\)

Exercises for Pascal’s Triangle

Solve.

1. Write the \(6th\) row of Pascal’s Triangle.
2. What is the sum of the \(12th\) row of Pascal’s triangle?

1. \(\color{blue}{1, 6, 15, 20, 15, 6, 1}\)
2. \(\color{blue}{4096}\)

Frequently Asked Questions

What is Pascal’s Triangle?

A triangular array of numbers where each interior entry is the sum of the two entries directly above it. The edges are all 1s.

How are the rows built?

Start with row 0: just 1. Row 1: 1 1. Row 2: 1 2 1. Row 3: 1 3 3 1. Row 4: 1 4 6 4 1. Each new row begins and ends with 1, and interior entries sum the two above.

What do the numbers represent?

The binomial coefficients C(n, k) = n!/(k!(n-k)!). Row n, position k gives C(n, k) — the number of ways to choose k items from n.

How is Pascal’s Triangle used in algebra?

To expand (a + b)^n. The coefficients of the expansion are row n of Pascal’s Triangle. For (a+b)^4: 1, 4, 6, 4, 1 — so (a+b)^4 = a^4 + 4a^3 b + 6 a^2 b^2 + 4 a b^3 + b^4.

Walk through expanding (a + b)^3.

Row 3 of Pascal’s Triangle: 1, 3, 3, 1. So (a + b)^3 = 1 a^3 + 3 a^2 b + 3 a b^2 + 1 b^3 = a^3 + 3 a^2 b + 3 a b^2 + b^3.

What is the sum of row n?

Exactly 2^n. Row 0 sums to 1, row 1 to 2, row 2 to 4, row 3 to 8, etc. This is because plugging a = b = 1 into (a+b)^n gives 2^n.

Are there patterns in the triangle?

Many. Fibonacci numbers appear along certain diagonals. Triangular numbers (1, 3, 6, 10, …) appear in another diagonal. Many number sequences hide inside Pascal’s Triangle.

How is it connected to probability?

Row n, position k gives the number of ways to get exactly k heads in n flips of a fair coin. Divided by 2^n, it gives the binomial probability.

Who was Pascal?

Blaise Pascal (1623-1662), a French mathematician, physicist, and philosopher. The triangle was known long before him (in China and Persia), but his work formalized many of its applications.

Where is Pascal’s Triangle used today?

Binomial expansions, combinatorics, probability theory, computer science algorithms (dynamic programming), and number theory.

Related Lessons You May Like

• How to solve permutations and combinations
• The binomial theorem
• Conditional and binomial probabilities
• How to find probability of an event
• Probability distribution

For deeper combinatorics and the binomial theorem, Algebra II for Beginners covers expansion and counting techniques. Pre-Calculus for Beginners extends into sequences and series.