# Thread: combination equation

1. ## combination equation

Solve in R :
n , p naturels numbers and

2. Observe that

$\displaystyle ^{n-1}\mathrm C_{p-1}\,+\,^{n-1}\mathrm C_p\ =\ ^n\mathrm C_p$

Hence

$\displaystyle x^2\,-\,^n\mathrm C_px\,+\,^{n-1}\mathrm C_{p-1}\cdot^{n-1}\mathrm C_p$

$\displaystyle =\ x^2\,-\,\left(^{n-1}\mathrm C_{p-1}\,+\,^{n-1}\mathrm C_p\right)x\,+\,^{n-1}\mathrm C_{p-1}\cdot^{n-1}\mathrm C_p$

$\displaystyle =\ \left(x\,-\,^{n-1}\mathrm C_{p-1}\right)\left(x\,-\,^{n-1}\mathrm C_p\right)$

3. Originally Posted by TheAbstractionist
Observe that
$\displaystyle ^{n-1}\mathrm C_{p-1}\,+\,^{n-1}\mathrm C_p\ =\ ^n\mathrm C_p$
Hence
$\displaystyle x^2\,-\,^n\mathrm C_px\,+\,^{n-1}\mathrm C_{p-1}\cdot^{n-1}\mathrm C_p$
$\displaystyle =\ x^2\,-\,\left(^{n-1}\mathrm C_{p-1}\,+\,^{n-1}\mathrm C_p\right)x\,+\,^{n-1}\mathrm C_{p-1}\cdot^{n-1}\mathrm C_p$

$\displaystyle =\ \left(x\,-\,^{n-1}\mathrm C_{p-1}\right)\left(x\,-\,^{n-1}\mathrm C_p\right)$
Hello : Thank you
it is necessary to discuss the number of resolution according to paramétres n , p