Question

Solve for n if nC2=nC4

Answer 1

n = 6

Explanation:

There's a secret formula for this:

If nCx = nCy,

Then n = x + y.

Hence, for this question, n = 2 + 4 = 6

Answer 2

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

`\boxed{\sf ^nC_r=(n!)/(r!(n-r)!)}`

`\\ \qquad\quad\sf\longmapsto (n!)/(2!(n-2)!)=(n!)/(4!(n-4)!)`

`\\ \qquad\quad\sf\longmapsto (1)/(2* 1(n-2)!)=(1)/(4* 3* 2* 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=(7)/(3)\pm (1)/(3)√(3)`