How to calculate Icon win for 4 Mirrors from the “Hexbreaker” Mirror Bonus?

I am reading the articles “Deconstructing Hexbreaker” and trying to calculate the probability and wins, but I have no idea how to calculate Icon win (Treasure symbol win) for 4 mirrors and 5 mirrors from the Mirror Mirror Bonus game.
I have tried to calculate these wins using Binomial tree but I could not get the same value.
Would anybody help me to find the right way to calculate this? Many thanks.

Each of the five treasure symbols has an equal chance of being behind any of the 13 mirrors. This means each symbol has a 5/13 chance of behind behind any of the five broken mirrors.

Let P(n) be the probability of n symbols being revealed behind broken mirrors.
P(5) = (5/13)⁵
P(4) = 5 x (5/13)⁴ x 8/13
P(3) = 10 x (5/13)³ x 8/13²
P(2) = 10 x (5/13)² x 8/13³
P(1) = 5 x 5/13 x 8/13⁴
P(0) = 8/13⁵

The expected payout for the 4-mirror bonus is P(5) x 1000 + P(4) x 125 + P(3) x 75 + P(2) x 25 + P(1) x 15 + P(0) x 0 = about 45.748237
(for those of you who want the exact number, it is 16,986,000 / 371,293)

Thank you for your detailed and clear explanation.

Hi ThatDonGuy,

Finally got it the result !! Many many thanks.
Still don't get the x5, x10, x10, x5 multipliers. Can anyone explain?

Sub Main()

Dim P(0 To 5) As Double
Dim R As Double
Dim i As Long
Dim SP As Double

'16986000 / 371293
' 5+8=13

P(5) = ((5 / 13) ^ 5) * ((8 / 13) ^ 0) * 1
P(4) = ((5 / 13) ^ 4) * ((8 / 13) ^ 1) * 5
P(3) = ((5 / 13) ^ 3) * ((8 / 13) ^ 2) * 10
P(2) = ((5 / 13) ^ 2) * ((8 / 13) ^ 3) * 10
P(1) = ((5 / 13) ^ 1) * ((8 / 13) ^ 4) * 5
P(0) = ((5 / 13) ^ 0) * ((8 / 13) ^ 5) * 1

For i = 0 To 5
Debug.Print "P(" & i & ") = " & Format(P(i), "##0.00000000")
SP = SP + P(i)
Next

Debug.Print "P = " & Format(SP, "##0.00000000")

R = P(5) * 1000 + P(4) * 125 + P(3) * 75 + P(2) * 25 + P(1) * 15 + P(0) * 0

Debug.Print R ' 45.7482365678858 !!

End Sub