The number of ways in which 13 gold coins can be distributed among three persons such that each one gets at least two gold coins is

36

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Solution

The correct option is A $36$
Formula $^{(n + r - 1)} C_{r - 1}$ ; This means you have to distribute n things among r people where anyone can get any number of things.
Since each should have atleast two coins, after giving 2 coins to each, we have $13 - 3 \times 2 = 7$ coins
left, which can be distributed among three.
This problem is equivalent to number of non negative integral solution of
$x_{1} + x_{2} + x_{3} = 7$ , which is
$^{(7 + 3 - 1)} C_{3 - 1} =^{9} C_{2} = 36$

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