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The Binomial Theorem and Pascal’s Triangle

February 24th, 2010 by Steven Pomeroy | Filed under Math Patterns, Math Tricks, Pascal's Triangle.




Back in grade school, I was first introduced to the Binomial Theorem.  The title alone was quite enough to intimidate me, let alone the seemingly impossible to understand equations involved with it.

I’ll not go into the mathematics of the binomial theorem here.  Instead, I’ll introduce you to math tricks which can be used instead.  First, let me refresh your mind on why we were taught the binomial theorem.  Remember when you were asked to expand the equation:

(x + y)2

If you recall, this equation can be expanded to the equivalent equation:

x2 + 2xy + y2

The binomial theorem will allow you to solve a higher order problem of the example above.  For instance, what is the expansion of the equation:

(x + y)5

Generally, an equation of this type can be expanded as:

binomial expansion

where c1, c2, … are the binomial coefficients in the expansion.   So given any n, you can determine the expansion without the coefficients.  Expanding our example above:

(x + y)5 = c1x5y0 + c2x4y1 + c3x3y2 + c4x2y3 + c5x1y4 + c6x0y5

So how do you determine the binomial coefficients?  You can determine the binomial coefficients individually using the equation:

binomial coefficients

for k=0 to k=n.  This works fine, but is a little bit cumbersome – especially for large values of n!  So what is the math trick to solve this quickly?

Before I can answer this, I have to introduce to you Pascal’s Triangle.  Pascal’s triangle is a mathematical progression which is determined by constructing a triangle with numbers using a very simple algorithm.  First, take a look at this example of Pascal’s triangle:

hexagonal pascal triangle

At the very top is row 0, which is simply a 1.  In row 1, there are two numbers, both 1s.  In row 2, there are three numbers: 1, 2, and 1.  Notice that the 2 in row two is the sum of the two numbers above it; this is how you determine the numbers in the triangle – simply add two side-by-side numbers to get the result below and between the numbers:

Animated Pascal TriangleConstruction of Pascal’s Triangle1

So how can you use Pascal’s triangle to find the binomial coefficients when you expand the equation (x + y)5?  First, notice that the equation is raised to the 5th power.  So now simply go to the 5th row of Pascal’s triangle (remember, the top row is row 0), and those numbers are the required coefficients:

1 5 10 10 5 1

And so,

(x + y)5 = x5y0 + 5x4y1 + 10x3y2 + 10x2y3 + 5x1y4 + x0y5

Quite a time saver!

 

For an alternative method of expanding polynomials, please check out The Easy Peasy Binomial Expansion Trick.

1File by Hersfold, en.wikipedia.org/wiki/User:Hersfold

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20 Responses to “The Binomial Theorem and Pascal’s Triangle”

  1. Miles | 11/08/10

    thatz wy im so hard to think.

  2. admin | 17/08/10

    A little work until the “ahh-haa!” moment . . .

  3. kres dee | 20/08/10

    ..midnight report that’s why i came rushing!

  4. admin | 26/08/10

    I hope it helped you out :D

  5. Darlene David | 4/09/10

    I have to solve the following problem using the form of the bionomial theorem using the first 2 terms. It is a bit confusing. Can you help?

    1/1.001 = 1 + Co

    1/0.999 = 1 + Co

  6. Rahul | 11/11/10

    you’re hot.

  7. admin | 11/11/10

    Why thanks!

  8. Mahesh Dharmawardane | 23/12/10

    Binomial theorem can be used for finding the square,cubic power,…, nth power of a given number.

  9. krystal | 3/10/11

    math is soooooooooooooo hard its like a whole diff lang. now i am stuck doing a all night report on binomial therom and some times i wish i was in pre-k where 2+2=4 was math not y=mx+b and stuff like that…………………………………………….bored

  10. Niacy | 31/10/11

    It really cool it help alot

  11. Nyssa | 8/11/11

    ummm yeah i get it to that point but what do you do past that. Ok I have the problem (a+3)”cubed”. How do I solve it, I only get it when theres nt real numbers involved.

  12. jessa mae | 5/03/12

    maka EYEBLEED….!

  13. រឿន | 16/07/12

    This is cool, it’s quite a time saver :-)

  14. old man | 22/08/12

    so is this right?
    3(x+y)**3=3(x**3) + 9(x**2)y + 9x(y**2) + 3y**3

  15. it IS beautiful! | in stillness the dancing | 1/09/12

    […] and look for others in Pascal’s Triangle.  In high school, the connection between the binomial theorem and Pascal’s Triangle are noted.  At various levels we explore the connection between […]

  16. Charles Lynch | 24/01/13

    You are the only one that helped me get that “AHA!” moment
    Test tomorrow.. thank you!! :)

  17. Math Tricks | 30/01/13

    I am very gratified Charles – very happy to help!! :)

  18. R,G, | 30/07/13

    This is not math. It is trick of the kind you teach dogs. If I want to expand a binomial, I use a computer and get

    >&expand((1+x)^20)

    20 19 18 17 16 15
    x + 20 x + 190 x + 1140 x + 4845 x + 15504 x
    14 13 12 11 10
    + 38760 x + 77520 x + 125970 x + 167960 x + 184756 x
    9 8 7 6 5 4
    + 167960 x + 125970 x + 77520 x + 38760 x + 15504 x + 4845 x
    3 2
    + 1140 x + 190 x + 20 x + 1

    in a fraction of a second.

    Math is to prove that the Pascal triangle get the coefficients of the expansion.

  19. trojdor | 5/05/14

    @R,G,

    R,G,
    You do realize that everything you study today involving math was discovered/invented long, long BEFORE your silly computer? Using a computer to think for you is the real “kind of trick you teach dogs”.

  20. Bill_from_SF | 26/07/16

    @R,G, — the other thing is, your “computer function” generates an incorrect result. The “computer result” always shows the exponent for x = 1. But actually 20 of the 21 terms in the polynomial expansion have non-zero and independent exponent values. So maybe best to follow the teacher and see how to generate the calculations! :-)

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