Concept: The question tests on the area of Modern Math (Counting/Combinatorics)
If the number of non-negative integer solutions for the equation x1+x2 + ..+ xn =n,then the number of ways the distribution can be done is = n+r−1Cr−1
In this case, value of any variable can be zero.
let us now look at the conditions.
=>We have that Shyam must have at least 2 chocolates and Ram can have a maximum of 5 chocolates. This means that Ram can have 0, 1, 2, 3, 4 or 5 chocolates.
Let us Give Shyam 2 chocolates, which takes out one condition.
We now have 13 chocolates. let Shyam + 2 = S', where S' can be 0 as it includes 2 chocolates.
When Ram has 0 chocolates, then S' + B + G = 13. n = 13 and r = 3.
n+r−1Cr−1 = 15C2 = 105
When Ram has 1 chocolates, then S' + B + G = 12. n = 12 and r = 3
When Ram has 2 chocolates, then S' + B + G = 11. n = 12 and r = 3.
When Ram has 3 chocolates, then S' + B + G = 10. n = 12 and r = 3.
When Ram has 4 chocolates, then S' + B + G = 9. n = 12 and r = 3.
When Ram has 5 chocolates, then S' + B + G = 8. n = 12 and r = 3.
Total number of ways = 105 + 91 + 78 + 66 + 55 + 45 = 440