I have N apples and M oranges and few random pairs of (N+M) children.
Distribute N apple and M orange among (N+M) children in such a way that, minimum numbers of pairs of children can have different fruits in a pair.
For better understanding lets take an example:
Ex 1) 2 apples and 1 orange and children=(2+1)=3
Given pairs of children are: (1,2) (2,3)
Now if i give an apple to 1 and 3 and orange to 2, then i will have two pairs with different fruits in a pair. i.e (apple,orange) (orange,apple)
But if i give an apple to 1 and 2 and orange to 3, then i will have only one pair with different fruits in a pair. i.e (apple,apple) (apple,orange).
So answer here is 1 pair which is minimum.
Note:
Number of pairs can be anything but >0 and <= nCr(n=no. of children and r=2)
Your problem is equal to the problem called graph partitioning.
Let every child be a vertex $v$ and every chosen pair be an edge $e$ in graph $G$.
To find apple and orange distribution between children is to find a partition of the graph vertex set $V(G)$ into 2 parts $V_1$ and $V_2$ such as $|V_1|=N$, $|V_2|=M$ and $V_1\cap V_2=\emptyset$,
while minimizing the number of edges between parts ( edges $(v,u)\in E : v \in V_1$ and $u \in V_2$ ).
The minimization of this edge number will minimize the number of pairs that have different fruits in it.
Also, the graph bisection problem which is known to be NP-hard can be reduced to your problem, so your problem is NP-hard too. (Proof: Let $N = M$)
So, if you want to find an algorithm for your problem you can use algorithms for graph partitioning. This survey can help you to find a better one for your needs.