Discrete Math: Urgent...can you help? In chemistry, the electrons in a given rin

Question

Discrete Math: Urgent...can you help?

In chemistry, the electrons in a given ring of an atom can each be in. one of several: energy states. How many ways can six electron to be assigned to five energy states? IT'S easy enough to think of electrons as balls and the states as Urns. The energy- states are distinguishable, because they correspond to different amounts of energy, and we can measure that. The harder question is: Do electrons act distinguishably? We think of them as distinct entities, so it is natural to suppose they will act distinguishably. However, it is an empirical fact that they don't. The appropriate model is putting indistinguishable balls in distinguishable urns, as in Example 2.

Explanation / Answer

We can take the problem as.
Find the total number of combinations of energy states from 5 different energy states(atleast 1 energy state is to be selected).
We can do this because, all energy states are different. And, any energy state can have any number of electrons.

So, number of selection of 1 energy state = 5C1 = 5
So, number of selection of 2 energy states = 5C2 = 10
So, number of selection of 3 energy states = 5C3 = 10
So, number of selection of 4 energy states = 5C4 = 5
So, number of selection of 5 energy states = 5C5 = 1

So, total number of ways in which any number of electrons can be assigned to 5 energy states = (5 + 10 + 10 + 5 + 1) = 31

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