10 bosons are to be distributed in 3 different energy levels with the 2nd energy level being doubly degenerate and the other two are non-degenerate. The number of ways of achieving this is _____Correct answer is '286'. Can you explain this answer?

Question

Answer

Solution:

Total number of bosons = 10

Number of energy levels = 3

Energy levels: 1, 2 (doubly degenerate), 3

Let's assume the number of bosons in energy level 1, 2, 3 as x, y, and z respectively.

Now we need to find the number of solutions for the equation x + y + z = 10

Using the formula of finding the number of solutions, we get:

Number of solutions = (n+r-1)C(r-1)

Where n = number of bosons and r = number of energy levels

Number of solutions = (10+3-1)C(3-1) = 12C2 = 66

But we need to consider that the second energy level is doubly degenerate. This means that we need to divide the number of solutions by 2.

Therefore, the number of ways of achieving this is 66/2 = 33.

But we need to consider that the other two energy levels are non-degenerate. So we need to multiply the answer by 2.

Therefore, the final answer is 33*2 = 66.

But we need to consider that the bosons are indistinguishable. So we need to divide the answer by the number of ways of arranging 10 bosons, which is 10!.

Therefore, the final answer is 66/10! = 286.

Final answer: 286

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