Prove that `r^1 = 1 + 2/n` where n is number of degrees of freedom.

The number of degree of freedom for diatomic gas is:

Prove that n! + (n + 1)! = n! (n + 2)

STATEMENT-1 : The average translational kinetic energy per molecule of the gas per degree of freedom is 1/2 KT. STATEMENT-2 : For every molecule there are three rotational degree of freedom.

Prove that n! (n+2)=n!+(n+1)! .

There are n straight lines in a plane such that n_(1) of them are parallel in onne direction, n_(2) are parallel in different direction and so on, n_(k) are parallel in another direction such that n_(1)+n_(2)+ . .+n_(k)=n . Also, no three of the given lines meet at a point. prove that the total number of points of intersection is (1)/(2){n^(2)-sum_(r=1)^(k)n_(r)^(2)} .

Prove that : ^nP_r= n"^(n-1)P_(r-1) , for all natural numbers n and r for which the symbols are defined.