Buns were being sold at three prices: One for a penny, two for a penny and three for a penny. Some children (there were as many boys as girls) were given seven pennies to spend on these buns, each child to receive exactly the same value in buns, assuming that all buns remained whole, how many buns and what type did each child receive?

I am working on this problem. So far I have come to this. Let x be the no of buns bought by each child. For each child, If a1 is the no of buns of first type, a2 the buns of second type, and a3 the no of buns of third type, then the total no of buns should be: \[a1+2a2+3a3=x\] Now, We should find the total no of solutions of this using binomial theorem. I will post as soon as I progress more

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