Combinatorics with Pascal's Triangle
Read this topicExplanations 8
- NOTE: read the other explanations to find out what a Binomial Coefficient is. In this explanation we will be studying something called Pascal's Triangle, and how it relates to b…
By Daniel Liu, Joshua Siktar
- Pascal's Triangle The math behind Pascal’s Triangle is very simple. It is so simple that even young students can learn and repeat the pattern. But as you look more closely, you …
By Trent Tormoehlen, Andrew Wang, Archit Kulkarni, Moor Xu, Joshua Siktar, Anusha Rahman
- Let's start with an example: (x+y)2=x2+2xy+y2(x+y)^2 = x^2 + 2xy + y^2(x+y)2=x2+2xy+y2 The coefficients (111, 222, and 111) happen to be the same as the second row of Pascal's T…
By Bladyda, Toan Pham, Ruiran Xun
- Pascal's Triangle, named after Blaise Pascal, is a triangle where two numbers added up, result in the next number: The top row of the triangle, containing only a single 111, is …
By Bladyda, Ruiran Xun, Anusha Rahman
- (Video) Binomial Expansion Using Pascal's Triangle by Mathispower4u This video by Mathispower4u demonstrates how to expand using Pascal's Triangle and the binomial theorem.
By Stefan Cuevas, Alex Federspiel, Ruiran Xun, Joshua Siktar
- প্যাসকেলের ত্রিভুজ কিছু সংখ্যার একটি ত্রিভুজাকার তালিকা। এই ত্রিভুজটি আকারে অসীম। নীচে ত্রিভুজটির 000 থেকে 101010 সারি দেখানো হলো। আচ্ছা, তাহলে এই ত্রিভুজের মানেটা কী? আর কেমনেই…
By Ittihad Ahmed · বাংলা
- Pascal’s Triangle is a triangle made of numbers. As the name suggests, the numbers are arranged in rows that grow longer as you move down the triangle. Each number in the triang…
By Ruiran Xun, Jonathan Heller, Anusha Rahman
- Background Before we discuss what exactly Pascal's triangle is, we briefly discuss what factorials and binomial coefficients are, because those are important for understanding w…
By Joshua Siktar, Anusha Rahman
Problems 24
- The sum of the entries in the 181818th row of the Pascal's Triangle can be written in the form aba^bab where aaa is a prime number. What is the value of a+ba+ba+b?
- What is the third term in row 141414 of Pascal's Triangle?
- What is the sum of the first 555 entries in row 999 of Pascal's Triangle?
- What is the sum of the first 141414 terms of the sequence 1+4+10+20+35+...1+4+10+20+35+...1+4+10+20+35+... where each term is the sum of the first nnn triangular numbers?
- What is the coefficient of the x5y2x^5y^2x5y2 term in the expansion of (x+y)7(x+y)^7(x+y)7?
- Expand (a+b)5(a+b)^5(a+b)5
- Expand: (a−4)3(a-4)^3(a−4)3
- What is the coefficient of the x5yx^5yx5y term in (2x+3y)6(2x+3y)^6(2x+3y)6?
- What is the sum of ALL of the entries of row 8 of Pascal's triangle? (Assume that the first row, a single 111, is row 0.)
- Compute (70)+(71)+(72)+⋯+(77)\binom70+\binom71+\binom72+\cdots+\binom77(07)+(17)+(27)+⋯+(77).
- The number 104,060,465104{,}060{,}465104,060,465 is divisible by a five-digit prime number. What is that prime number? (Hint: Do the digits look like a row of Pascal's triangle?…
- What is the largest number in the tenth row of Pascal's Triangle? (Hint: You don't have to write out the whole triangle to figure this out. Use combinatorics!)
- What is the 512512512th number in the 512512512th row of Pascal's Triangle?
- What is the third number in the 364364364th row of Pascal's Triangle? Write your answer without commas or spaces. (Hint: Find a pattern!)
- What does the horizontal symmetry of Pascal's Triangle tell us about combinatorics?
- Find the number of positive integers that are less than or equal to 200320032003 and whose base-222 representation has more 111's than 000's. Give the last three digits of your …
- What is the sum of the elements in the 111111th row of Pascal's triangle?
- How can we write the 999th element of the 212121st row of Pascal's triangle as a binomial coefficient?
- How is the third element of the eighth row of Pascal's triangle expressed as a binomial coefficient?
- In the binomial expansion of (x+2y)4(x+2y)^4(x+2y)4, what is the coefficient of the xy3xy^3xy3 term?
- What is the sum of all elements in the 111111th row of Pascal's triangle?
- What is the sum of all elements in the 666th row of Pascal's triangle?
- What is (75)?{{7}\choose{5}}?(57)?
- We want to compute some binomial coefficients knowing the values (50)=1{{5}\choose{0}} = 1(05)=1 (51)=5{{5}\choose{1}} = 5(15)=5 (52)=10{{5}\choose{2}} = 10(25)=10 (53)=10{{5…