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Pascal's triangle

Pascal's triangle is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Pascal's triangle rather than just read about it. In short: In mathematics, Pascal's triangle is an infinite triangular array of the binomial coefficients which play a crucial role in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although other mathematicians studied it centuries before him in India, Persia, China, Germany, and Italy.

Pascal's triangle — main illustration
Pascal's triangle — illustration

Key takeaways

  • Pascal's triangle belongs to mathematics; place it in that map before memorising details.
  • Learn the definition first, then one example that makes the definition concrete.
  • Connect Pascal's triangle to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Pascal's triangle from memory before moving on to harder problems.

Reference excerpt

In mathematics, Pascal's triangle is an infinite triangular array of the binomial coefficients which play a crucial role in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although other mathematicians studied it centuries before him in India, Persia, China, Germany, and Italy. The rows of Pascal's triangle are conventionally enumerated starting with row n = 0 {\displaystyle n=0} at the top (the 0th row). The entries in each row are numbered from the left beginning with k = 0 {\displaystyle k=0} and are usually staggered relative to the numbers in the adjacent rows. The triangle may be constructed in the following manner: In row 0 (the topmost row), there is a unique nonzero entry 1. Each entry of each subsequent row is constructed by adding the number above and to the left with the number above and to the right, treating blank entries as 0. For example, the initial number of row 1 (or any other row) is 1 (the sum of 0 and 1), whereas the numbers 1 and 3 in row 3 are added to produce the number 4 in row 4.

Formula In the n {\displaystyle n} th row of Pascal's triangle, the k {\displaystyle k} th entry is denoted ( n k ) {\displaystyle {\tbinom {n}{k}}} , pronounced "n choose k" because it describes the number of combinations: the number of ways of choosing ⁠ k {\displaystyle k} ⁠ things from among a collection of ⁠ n {\displaystyle n} ⁠ things. Row numbering starts at 0, and likewise entries within a row are numbered from 0. For example, the topmost entry is ( 0 0 ) = 1 {\displaystyle {\tbinom {0}{0}}=1} . With this notation, the construction of the previous paragraph may be written as

( n k ) = ( n − 1 k − 1 ) + ( n − 1 k ) {\displaystyle {n \choose k}={n-1 \choose k-1}+{n-1 \choose k}} for any positive integer n {\displaystyle n} and any integer 0 ≤ k ≤ n {\displaystyle 0\leq k\leq n} . This recurrence for the binomial coefficients is known as Pascal's rule. An arbitrary binomial coefficient can be calculated as

( n k ) = n ! k ! ( n − k ) ! . {\displaystyle {n \choose k}={\frac {n!}{k!(n-k)!}}.}

History

The pattern of numbers that forms Pascal's triangle was known well before Pascal's time.

… excerpt ends here. Continue reading the full article.

Illustrations

Pascal's triangle: Yang Hui's triangle, as depicted by the Chinese using rod numerals, appears in Jade Mirror of the Four Unknowns, a mathematical work by Zhu Shijie, dated 1303.
Yang Hui's triangle, as depicted by the Chinese using rod numerals, appears in Jade Mirror of the Four Unknowns, a mathematical work by Zhu Shijie, dated 1303.
Pascal's triangle: Pascal's version of the triangle
Pascal's version of the triangle
Pascal's triangle: Visualisation of binomial expansion up to the 4th power
Visualisation of binomial expansion up to the 4th power
Pascal's triangle: Each frame represents a row in Pascal's triangle. Each column of pixels is a number in binary with the least significant bit at the bottom. Light pixels represent 1 and dark pixels 0.
Each frame represents a row in Pascal's triangle. Each column of pixels is a number in binary with the least significant bit at the bottom. Light pixels represent 1 and dark pixels 0.
Pascal's triangle: The numbers of compositions of n+1 into k+1 ordered partitions form Pascal's triangle.
The numbers of compositions of n+1 into k+1 ordered partitions form Pascal's triangle.

Worked examples

Example 1 — a first encounter with Pascal's triangle

Start with the simplest possible case. Write down what Pascal's triangle claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Pascal's triangle before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Pascal's triangle ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Pascal's triangle

In research
Pascal's triangle appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry
Engineering practice reuses Pascal's triangle in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom
Pascal's triangle is common in secondary-school and first-year university syllabi. It links to neighbouring topics Blaise Pascal, Factorial and binomial topics, Triangles named after people, so understanding it makes those chapters shorter.
In everyday life
Look for Pascal's triangle outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.
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How to study Pascal's triangle in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Pascal's triangle means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Pascal's triangle out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Pascal's triangle in simple terms?

In mathematics, Pascal's triangle is an infinite triangular array of the binomial coefficients which play a crucial role in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although other mathematicians studied i…

Why does Pascal's triangle matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Pascal's triangle?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Pascal's triangle.

Tags

  • Blaise Pascal
  • Factorial and binomial topics
  • Triangles named after people
  • Triangles of numbers

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