What is Pascal's triangle: Definition and 22 Discussions

In mathematics, Pascal's triangle is a triangular array of the binomial coefficients that arises 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


{\displaystyle n=0}
at the top (the 0th row). The entries in each row are numbered from the left beginning with


{\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 in the first (or any other) row is 1 (the sum of 0 and 1), whereas the numbers 1 and 3 in the third row are added to produce the number 4 in the fourth row.

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  1. S

    I Sum of Binomial Expansion | Spivak Chapter 2, Excercise 3 part d

    Hello, I am working through Spivak for self study and sharpening my math skills. I have become stuck on an exercise. What I need to show is the following: $$ (a + b) \sum_{j = 0}^{n} \binom nj a^{n-j}b^{j} = \sum_{j = 0}^{n + 1} \binom{n+1}{j} a^{n-j + 1}b^{j} $$ My attempt, starting from...
  2. HorseRidingTic

    Quadratics using Pascal's triangle

    Homework Statement The problem equation is contained in the picture. Homework Equations Pascal's Triangle is useful is this one. The Attempt at a Solution The difficulty I'm having is in going between lines 2 and 3 which I've marked with a little red dot. The closest I get to simplifying...
  3. J

    A Is this product always greater than these sums?

    I've been working on a problem for a couple of days now and I wanted to see if anyone here had an idea whether this was already proven or where I could find some guidance. I feel this problem is connected to the multinomial theorem but the multinomial theorem is not really what I need . Perhaps...
  4. G

    What is the equation for predicting the middle number in Pascal's triangle?

    I am trying to find the equation to predict the next middle number in pascal's triangle. By middle number I mean in each row that has odd number of numbers the middle number of that row. So for example row 6 which has 1,6,15,20( middle number), 15,6,1. I am trying to find that middle number, but...
  5. craigi

    Many Worlds, Indistinguishability, Pascal's Triangle and the Gaussian

    Suppose we have a quantity which can take discrete equally spaced values. Iteratively, we can increase or decrease this quantity by one quantum, splitting into two new worlds each time. After multiple iterations we have some indistinguishable worlds, as described by Pascal's triangle. As the...
  6. Seydlitz

    Proving natural numbers in Pascal's Triangle

    Homework Statement Taken from Spivak's Calculus, Prologue Chapter, P.28 b) Notice that all numbers in Pascal's Triangle are natural numbers, use part (a) to prove by induction that ##\binom{n}{k}## is always a natural number. (Your proof by induction will be be summed up by Pascal's...
  7. L

    Connection between polynomials and Pascal's triangle

    I recently discovered that for a 3rd degree polynomial I was studying, f(5) - 4f(4) + 6f(3) - 4f(2) + f(1) = 0. At first I just though it was coincidental that the coefficients were the 5th row of Pascal's Triangle, but then I tried a 2nd degree polynomial and found that f(4) - 3f(3) + 3f(2) -...
  8. T

    Number of unique paths in Pascal's Triangle

    Homework Statement Let pascal(n,i) be the value of the ith element of the nth row of Pascal's triangle. Using induction show that the number of unique paths from entry {0,0} to entry {n,i} in Pascal's triangle is equal to pascal(n,i). The Attempt at a Solution The base case n=1 seems easy...
  9. S

    Formula for the n-th row of Pascal's Triangle

    Homework Statement Find a formula for the sum of the elements of the nth row of Pascal's Triangle Homework Equations C(n,r) = [C(n-1,r-1) + C(n-1,r)] C(n,0) = C(n,n) = 1 The Attempt at a Solution I started with the summation of the elements in the rows n \sum^{n}_{r=0} C(n,r)...
  10. K

    Prove that the rᵗʰ term in the nᵗʰ row of Pascal's triangle is nCr.

    Prove that the rᵗʰ term in the nᵗʰ row of Pascal's triangle is nCr. nCr formula: n!/r!(n-r)! I've tried everything I can but I don't know how to approach this question.
  11. S

    Calculating Routes on a Grid: Using Combinations and Pascal's Triangle

    1. The streets of a city are laid out in a rectangular gird, as shown below a) Use combinations to find the number of routes through the grid that lead from point A to point B by only traveling north or east. Show your calculations b) How many of these routes pass through intersections...
  12. K

    Pascal's Triangle: 40 Paths to Spell BINOMIAL

    Homework Statement How many different paths will spell the word BINOMIAL in the following arrangement(moving diagonally downwards to the left or right)? ...B ...I I ..N N N O O O O .M M M ...I I ...A ...L L Homework Equations The Attempt at a Solution Starting from B using Pascal's Triangle...
  13. J

    Using pascal's triangle and standard form

    Homework Statement If z = 4 + i and w = -2 +2i, determine (z+w)5 using Pascal’s Triangle and standard form. Homework Equations The Attempt at a Solution
  14. Z

    Link between Pascal's triangle and integrals of trigonometric functions

    My lecturer keeps simplifying trigonometric integrals in one line such as \int^{2\pi}_{0}sin^{4}(t)dt=\frac{3\pi}{4} and writes pascals triangle next to it. Just wondering what's the link between them? I'm sure it's obvious and easy, I'd just like to have an fast way of dealing with these
  15. C

    Pascal's Triangle related question

    Let n and k be positive integers. After calculating several examples, guess a closed formula for: (n \ 0) + (n + 1 \ 1) + ... + (n + k \ k) If it helps, this is the formula for the sum of the nth row of the pascal triangle: (n \ 0) + (n \ 1) + ... (n \ k) = 2^n (n \ 0) means n...
  16. T

    Why does coupling intensity in H-NMR correspond to Pascal's triangle?

    All right, my 1st-year college chemistry class is just beginning NMR, and I really have no clue what's going on. But what caught my eye was how the relative intensities in hydrogen coupling is roughly predicted by Pascal's triangle. Is this because of probability? For a quartet, the number...
  17. S

    Permutations & Pascal's Triangle

    Homework Statement Water is poured into the top bucket of a triangular stack of 2-L buckets. When each bucket is full, the water overflows equally on both sides into the buckets immediately below. How much water will have been poured into the top bucket when at least one of the buckets in the...
  18. Pythagorean

    Pascal's Triangle for non-commutative?

    I was curious if there was a non-commutative version of Pascal's triangle for operators (such as those used in brah-ket notation) The important note is that (a + b)^2 = a^2 + b^2 + ab +ba where ba != ab
  19. S

    Pascal's Triangle and Binomial Theorem

    1. Evaluate the numbers for the coefficient of x4y9 in the expansion of (3x + y)13. 2. The Binomial Theorem states that for every positive integer n, (x + y)n = C(n,0)xn + C(n,1)xn-1y + ... + C(n,n-1)xyn-1 + C(n,n)yn. 3. I understand that the coefficients can be found from the n row of...
  20. S

    Question on a formula for Pascal's Triangle

    I'm doing an investigation on Binomial Coefficients in my HL math class, and the problem reads: "There is a formula connecting any (k+1) successive coefficients in the nth row of the Pascal Triangle with a coefficient in the (n+k)th row. Find this formula." Basically, what I did first was...
  21. A

    Binomial coefficients and pascal's triangle

    I am working through a mathematics olympiad problem book, and I am asked to prove that n choose r, where n is the row number and r is the term number in the row is equal to that term. Can someone please give me a hint? I have not been able to find ANY proofs on the internet through a basic...
  22. V

    Pascal's Triangle and Geometry

    Does anyone here know of any lengthy texts on the subject? I have been hard-pressed to find anything past the fact that the nth row can be used to construct an (n-2)-dimension simplex.