Binomial Coefficient
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.