Binomial Coefficient

Calculates the binomial coefficient ([n], [k]) \text{ }(n >= k)

This online tool calculates the binomial coefficient which can be useful in combinatorics to calculate the number of combinations. It is also used in the binomial formula to calculate polynomial coefficients of (a+b)^n polynom.

Definition of the binomial coefficient

Let n be a positive integer and k a positive integer less than or equal to n, then the binomial coefficient is the number of subsets of k elements choosen from the n elements (abbréviated to "n choose k"). The binomial coefficient is given as,

([n], [k]) = C_n^k = frac {n!}{k!(n-k)

n! denotes factorial n,

n! = n* (n-1) * (n-2)... *2*1

Newton binomial and binomial coefficient

Binomial coefficients are used to calculate the coefficients of a polynomial raised to a power n.

Example: calculate the coefficient of $$x^4y^2$$ in the expansion of $$(x + y) ^6$$

$$\text {coef} (x^4y^2) =\dbinom {6} {4} =\dfrac {6!} {4! \, 2!} = 15$$,

We'd calculate all the coefficients rather than expanding $$(x + y) ^6$$ which can be tedious!

Other coefficients:

$$\text {coef} (x^6) =\dbinom {6} {6} =\dfrac {6!} {6! \, 0!} = 1$$
$$\text {coef} (x^5y) =\dbinom {6} {5} =\dfrac {6!} {5! \, 1!} = 6$$
$$\text {coef} (x^4y^2) =\dbinom {6} {4} =\dfrac {6!} {4! \, 2!} = 15$$ (already calculated above)
$$\text {coef} (x^3y^3) =\dbinom {6} {3} =\dfrac {6!} {3! \, 3!} = 20$$
$$\text {coef} (x^2y^4) =\dbinom {6} {2} =\dfrac {6!} {2! \, 4!} = 15$$
$$\text {coef} (xy^5) =\dbinom {6} {1} =\dfrac {6!} {1! \, 5!} = 6$$
$$\text {coef} (y^6) =\dbinom {6} {0} =\dfrac {6!} {0! \, 6!} = 1$$

Therefore,

$$(x+y)^6=x^6+6x^5y+15x^4y^2+20x^3y^3+15x^2y^4+6xy^5+y^6$$

Notice that some coefficients are equal : $$x^2y^4$$ and $$x^4y^2$$ , $$xy^5$$ and $$x^5y$$. This symmetry is easily deduced from the binomial coefficient formula. This simplifies the calculations. Applied to $$(x+y) ^8$$ polynom, we get,
$$\text {coef} (x^8) = \text {coef} (y^8) = 1$$

For the other coefficients, we only calculate the coefficients of $$xy^7$$,$$x^2y^6$$,$$x^3y^5$$ and $$x^4y^4$$. The other coefficients are deduced symetrically.