elements from a set of elements. We can deduce the numeric value of ( choose ) using the formula:
! ! ( − )!
(
) =
Now have a look at the pattern to the right – we have formed Pascal’s Triangle. Using this method, we can create a bridge to better understand the relationship between Binomial Theorem and Pascal’s Triangle.
Therefore, according to what’s shown above, for (x + y) n , we can write down an expansion of:
0
1
2 ) −2 2 + ⋯+ (
− 1
( + ) = (
) 0 + (
) −1 1 + (
) 1 −1 + (
) 0
Or:
( + ) = ∑( 𝑘 ) =0
−
hence binomial theorem.
If we expand the binomial theorem further into multinomial theorem it’s going to become more interesting. We can first have a look at the example ( + + ) . As it is three- dimensional, we can visualize this with a matching Pascal’s Pyramid. And if we go into factorizing, it can be represented as:
( + + ) = ∑ −− ,
What if we have, say, subjects in the brackets and visualizing polygons can no longer support us? Since everything follows a pattern, there is a formula for multinomial theorem:
) = ∑ (
( 1
+ 2
+ ⋯+
)∏
𝑘 1
, 𝑘 2
, … , 𝑘
1 + 2 +⋯+ =
=1
Going through the full proof would take a very long time. There are articles you can find online that explain the theorem step by step – it only adds
another loop of getting a product on top of the coefficient calculation.
Now, on to some other sequences, there are a lot to find in this triangle and I will go through some briefly. Say we take all the odd numbers in a Pascal’s Triangle and leave the rest
4
Made with FlippingBook - professional solution for displaying marketing and sales documents online