26 - Binomial Discrete Mathematics Andrei Bulatov Discrete...

Info icon This preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon
Introduction Binomial Coefficients Discrete Mathematics Andrei Bulatov
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Discrete Mathematics – Binomial Coefficients 26-2 Previous Lecture Combinations with repetitions C(n + r – 1,r – 1) Also recall that )! ( ! ! ) , ( r n r n r n C r n - = =
Image of page 2
Discrete Mathematics – Binomial Coefficients 26-3 A Binomial A binomial is simply the sum of two terms, such as x + y We are to determine the expansion of Let us start with Every term in the expansion is obtained as the product of a term from the first binomial, a term from the second binomial, and a term from the third binomial Each of the terms xxy, xyx, and yxx is obtained by selecting y from one of the 3 binomials. Therefore, the coefficient 3 of is, actually, the number of 1-combinations from a set with 3 elements n y x ) ( + 3 y x ) ( + ) ( ) ( ) ( ) ( y x y x y x y x 3 + + + = + xyy xyx xxy yxx xyy xyx xxy xxx = 3 2 2 3 3 3 y xy y x x + + + = y x 2
Image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Discrete Mathematics – Binomial Coefficients 26-4 The Binomial Theorem Theorem . Let x and y be variables, and let n be a nonnegative integer. Then Proof. The terms in the product when it is expanded are of the form for j = 0,1,2,…, n. To count the number of terms of the form , note that to obtain such a term it is necessary to choose j y’s from the n binomials (so that the other n – j terms in the product are x’s).
Image of page 4
Image of page 5
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern