Now the number

**9**and

**The Final Question**

**9.**

The integral $Z(0)=\int^{\infty}_{-\infty} dx e^{-x^2}= \sqrt{\pi}$

**(a)**(3 POINTS:)Show that the integral $Z(j)=\int^{\infty}_{-\infty} dx e^{-x^{2}+jx}$

Where $j$ is not a function of $x$,is $Z(j)=e^{j^{2}/4a} Z(0)$

**(b)**(10 POINTS):Show that,

$\dfrac 1 {Z(0)}=\int x^{2n} e^{-x^2}= \dfrac {(2n-1)!!}{2^n}$

Where $(2n-1)!!$ is defined as $(2n-1)(2n-3)\times...\times3\times 1$

**(c)**(7 POINTS):What is the number of ways to form $n$ pairs from $2n$ distinct objects?Interept the

**previous part**of the problem in term of this answer.

[It was a

**number exam,and this is one of the toughest problems.]**

*200*