Solve the inequality `.^(x-1)C_(4)-.^(x-1)C_(3)-(5)/(2).^(x-2)C_(2)lt0,x inN`.

Explanation

We have, `.^(x-1)C_(4)-.^(x-1)C_(3)-(5)/(4).^(x-2)A_(2) lt 0`
`hArr((x-1)(x-2)(x-3)(x-4))/(1*2*3*4)-((x-1)(x-2)(x-3))/(1*2*3)`
`-(5)/(4)*(x-2)(x-3)lt0`
`hArr(x-1)(x-2)(x-3)(x-4)-4(x-1)(x-2)(x-3)-30(x-2)(x-3)lt0`
`hArr(x-2)(x-3){(x-1)(x-4)-4(x-2)-30}lt0`

`hArr(x-2)(x-3){x^(2)-9x-22}lt0`
`hArr(x-2)(x-3)(x+2)(x-11)lt0`
From wavy curve method
`x in(-2,2)cup(3,11)`
but `xinN`
`thereforex=1,4,5,6,7,8,9,10` . . .(i)
From inequality,
`x-1ge4,x-1ge3,x-2ge2`
or `xge5,xge4,xge4`
Hence, `xge5` . . . (ii)
From eqs.(i) and (ii) and solutions of the inequality are
`x=5,6,7,8,9,10`.