Question

The following problem arises in quantum mechanics (see Chapter $13$, Problem$7.21$). Find the number of ordered triples of nonnegative integers a, b, c whose sum$a+b+c$ is a given positive integer n. (For example, if$n=2$, we could have$(a,b,c)=(2,0,0)$or$(2,0,2)$or $(0,0,2)$or $(0,1,1)$or or $(1,0,1)$or $(1,1,0)$.) Hint: Show that this is the same as the number of distinguishable distributions of n identical balls in$3$boxes, and follow the method of the diagram in Example $5$.

Short answer

The required value is given below.

$M=C(n+2,n)$

Step-by-step solution

Given Information

Non negative integers whose sum is $a+b+c$.

Definition of uniform sample spaces

If a given experiment's sample space is known to be uniform, the probability of an event can be calculated using the event sizes and the sample space.

Find the values

Let the sum be $n=a+b+c$.

Bose-Einstein techniques in distribution is given below.

$M=C(R-1+K,K)$

Where K is number of balls or total sum of variable.

R is the number of variable or number of boxes.

$\begin{array}{c}M=C(3-1+n,n)\\ =C(n+2,n)\end{array}$

Hence the required value is mentioned below.

$M=C(n+2,n)$