Question

1) A\(^{3_{ }}n\) +5 . A\(^2_n\) = 2 . (n + 15) 
2) P\(_2\) . x - P\(_3\) . x = 8
3) A\(^3_x\) + 5 . A\(^2_x\) = 2 . ( x + 15 ) 
4) P\(_2\) . x\(^2\) - P\(_3\) . x = 8 
5) 3 . A\(^x_2\) - A\(^{2x}_2\) + 42 = 0 

Answer 1

1: \(\Leftrightarrow n\left(n-1\right)\left(n-2\right)+5\cdot n\left(n-1\right)=2n+30\)

=>n^3-3n^2+2n+5n^2-5n-2n-30=0

=>n^3+2n^2-5n-30=0

=>n=3

2: \(\Leftrightarrow2!\cdot x-3!\cdot x=8\)

=>-2x=8

=>x=-4

3: \(\Leftrightarrow\dfrac{x!}{\left(x-3\right)!}+5\cdot\dfrac{x!}{\left(x-2\right)!}=2x+30\)

=>\(x\left(x-1\right)\left(x-2\right)\)+5x(x-1)=2x+30

=>x=3

4: =>2x^2-6x=8

=>x^2-3x=4

=>x^2-3x-4=0

=>(x-4)(x+1)=0

=>x-4=0

=>x=4