What is the natural number x if C(x,2) + P(x,2) = 30?

C(x, 2) = x!/ ( 2! *(x -
2)!)

P(x , 2) = x! / (x -
2)!

Now C(x,2) + P(x,2) =
30

=> x!/ ( 2! *(x - 2)!) + x! / (x - 2)! =
30

=> x*(x - 1)/2! + x*(x - 1) =
30

=> x*(x - 1)/2 + x*(x - 1) =
30

=> x*(x - 1) + 2*x*(x - 1) =
60

=> x^2 - x + 2x^2 - 2x =
60

=> 3x^2 - 3x =
60

=> x^2 - x =
20

=> x^2 - 5x + 4x - 20 =
0

=> x(x - 5) + 4(x - 5) =
0

=> (x + 4)(x - 5)
=0

x can be -4 or 5.

A
negative value of x has no meaning , therefore it is ignored. So x =
5.

The required value of x =
5.

check:

C( 5,
2) + P(5 , 2) = 30.