Question

Solve for n if nC2=nC4

Answer 1

$\\ \qquad\quad\sf\longmapsto ^nC_2=^nC_4$

$\boxed{\sf ^nC_r=\dfrac{n!}{r!(n-r)!}}$

$\\ \qquad\quad\sf\longmapsto \dfrac{n!}{2!(n-2)!}=\dfrac{n!}{4!(n-4)!}$

$\\ \qquad\quad\sf\longmapsto \dfrac{1}{2\times 1(n-2)!}=\dfrac{1}{4\times 3\times 2\times 1(n-4)(n-3)(n-2)!}$

$\\ \qquad\quad\sf\longmapsto 2=24(n-4)(n-3)$

$\\ \qquad\quad\sf\longmapsto 2=24\left\{n(n-3)-4(n-3)\right\}$

$\\ \qquad\quad\sf\longmapsto 2=24(n^2-3n-4n+12)$

$\\ \qquad\quad\sf\longmapsto 2=24(n^2-7n+12)$

$\\ \qquad\quad\sf\longmapsto 2=24n^2-168n+288$

$\\ \qquad\quad\sf\longmapsto 24n^2-168n=-286$

$\\ \qquad\quad\sf\longmapsto 24n^2-168n+286=0$

$\\ \qquad\quad\sf\longmapsto 12n^2-84n+143=0$

$\\ \qquad\quad\bf\longmapsto n=\dfrac{7}{3}\pm \dfrac{1}{3}\sqrt{3}$