Question

We based the exact probabilities of equation (9.9) on the claim that the number of ways of adding \(N\) distinct nonnegative integers/quuntum numbers to give a total of \(M\) is \(\\{M+N-1) ! /\left[M^{\prime}(N-1) !\right]\). Verify this claim (a) for the case \(N=2, M=5\) and (b) for the case \(N=5, M=2\)

Short answer

The claim has been successfully verified with 6 distinct ways in Case (a) where \(N=2, M=5\). And in Case (b) where \(N=5, M=2\), there are 15 distinct ways.

Step-by-step solution

Analyzing function for \(N=2, M=5\)

By plugging the values \(N=2\) and \(M=5\) into the provided equation, we get: \(\frac{(5+2-1)!}{5!(2-1)!} = \frac{6!}{5!} = 6\). Now let’s reason this using combinatorics. \(N\) distinct nonnegative integers sum up to \(M\), in this case there are two numbers summing up to five. Those numbers can be (0,5), (1,4), (2,3), (3,2), (4,1), (5,0), a total of 6 ways.

Analyzing function for \(N=5, M=2\)

Now plug in the values for \(N=5\) and \(M=2\). We get: \(\frac{(5+2-1)!}{2!(5-1)!} = \frac{6!}{2!4!} = 15\). Reasoning this out using combinatorics. Now, we have five numbers sum up to two. Those numbers can be (0,0,0,0,2), (0,0,0,1,1), (0,0,0,2,0), (0,0,1,0,1), (0,0,1,1,0), (0,0,2,0,0), (0,1,0,0,1), (0,1,0,1,0), (0,1,1,0,0), (0,2,0,0,0), (1,0,0,0,1), (1,0,0,1,0), (1,0,1,0,0), (1,1,0,0,0), (2,0,0,0,0), which shows that there are 15 distinct ways the numbers can add up to two.

Key concepts

Permutations and Combinations

Understanding permutations and combinations is essential for solving problems that involve arranging or selecting objects. The two concepts hinge on counting the number of ways things can be organized.

Firstly, permutations refer to the arrangements of objects where the order does matter. For example, consider the ordering of books on a shelf; 'A-B-C' is different from 'C-A-B' because the order of the books differs.

On the other hand, combinations refer to the selection of objects where the order does not matter. Imagine choosing two desserts from a menu; selecting 'cake and ice cream' is the same as 'ice cream and cake' because the order of selection isn’t important.

In our exercise, considering the case where we have 2 distinct nonnegative integers that sum up to 5, we're dealing with combinations. Here, we are interested in the number of ways to select the integers without regard to order. We can use the combination formula which involves factorials, to find that there are \(6\) ways to combine two numbers to sum up to 5.

Factorial

The concept of factorial, denoted by an exclamation mark \( ! \), is central to many combinatorial calculations. It's defined as the product of all positive integers up to a given number. For example, \( 4! = 4 \times 3 \times 2 \times 1 = 24 \).

Factorials increase very rapidly with the number to which they are applied. They are incredibly useful for calculating permutations and combinations because they provide a way to describe the total number of possibilities in a compact form. In the context of our example, we used factorial to calculate the number of ways two numbers could sum up to 5 (\(6! / 5!\)) and the number of ways five numbers could sum up to 2 (\(6! / (2! \times 4!)\)).

Factorials also embody the principle of the multiplication rule in counting. In our scenarios, the factorial represents every possible arrangement of adding together numbers to reach total sums of 5 and 2, respectively.

Quantum Numbers

Quantum numbers are a fundamental aspect of quantum mechanics, which describe the unique quantum state of a particle. These numbers are not about counting and combinations in the traditional sense but rather signify the allowed values that describe the energy, angular momentum, magnetic moment, and spin orientation of an electron within an atom.

The term 'quantum numbers' used in the problem statement is a bit of a flourish, indicating distinct integer values. In physics, quantum numbers are discretized due to the wave-like nature of particles at small scales. However, the combinatorial problem of adding distinct nonnegative integers to reach a total sum can be likened to distributing units of energy among different energy levels while complying with certain rules.

To 'verify' claims in a combinatorial context, we apply the principles of permutations and combinations. Yet, for quantum numbers in physics, verification would involve experiments or complex calculations conforming to the laws of quantum mechanics, distinct from the simple mathematical reasoning in the earlier steps.