# Summing up binomial coefficients

• utkarshakash
In summary, the conversation discusses the value of a mathematical expression containing binomial coefficients and the use of properties and patterns to simplify the expression. The conversation also suggests considering other linear combinations for further simplification.
utkarshakash
Gold Member

## Homework Statement

The value of $((^n C_0+^nC_3+...) - \frac{1}{2} (^nC_1+^nC_2+^nC_4+^nC_5+...))^2 + \frac{3}{4} (^nC_1-^nC_2+^nC_4-^nC_5...)^2$

## The Attempt at a Solution

I can see that in the left parenthesis, the first bracket contains terms which are multiples of 3 and in the second bracket, those terms are missing. I know it's a much less attempt from my part but that's all I can see. Please help XO

Sums go to n?
Have you tried expanding out the squares?
You'll need extra binomial coefficients.

The coefficients have a symmetry amongst other properties - you should make a cheat-sheet of the basic properties.

Also - try n=2 and n=3 and expand them out explicitly and see if you find a pattern.

BTW: what was the question?

Try using powers of a cube root of 1. E.g. look at how the imaginary parts of those change and compare it with the behaviour of the last series.

haruspex said:
Try using powers of a cube root of 1. E.g. look at how the imaginary parts of those change and compare it with the behaviour of the last series.

I could simplify the first series to
'$\dfrac{(\omega ^2n + \omega ^n)^2}{4}$

But I still can't get the second one. Here's my attempt

$(1+\omega)^n + (1+ \omega ^2)^n = 2(C_0+C_3+...) - (C_1+C_2+C_4...)$

utkarshakash said:
I could simplify the first series to
'$\dfrac{(\omega ^2n + \omega ^n)^2}{4}$

But I still can't get the second one. Here's my attempt

$(1+\omega)^n + (1+ \omega ^2)^n = 2(C_0+C_3+...) - (C_1+C_2+C_4...)$

Consider other linear combinations of (1+1)n, (1+ω)n, (1+ω2)n.

## 1. What are binomial coefficients?

Binomial coefficients are a mathematical concept used to represent the coefficients of the terms in the expansion of a binomial expression. They are also known as Pascal's triangle coefficients.

## 2. How do you calculate binomial coefficients?

Binomial coefficients can be calculated using the formula nCk = n! / (k! * (n-k)!), where n is the total number of items and k is the number of items being chosen. Alternatively, they can also be found using Pascal's triangle.

## 3. What is the significance of binomial coefficients?

Binomial coefficients have many applications in mathematics, statistics, and computer science. They are used in probability calculations, combinatorics, and in the development of algorithms.

## 4. Can binomial coefficients be negative?

No, binomial coefficients are always positive integers. This is because they represent the number of ways to choose a certain number of items from a larger set, which cannot be negative.

## 5. What is the relationship between binomial coefficients and the binomial theorem?

The binomial coefficients are the coefficients of the terms in the binomial expansion, as given by the binomial theorem. This theorem states that (a + b)^n = Σ(nCk * a^(n-k) * b^k) for any positive integer n, where k ranges from 0 to n.

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