Answer:

We have two problems to solve: distributing 7 apples among 4 children so that each child gets at least 1; and distributing 6 oranges among 4 children with no restrictions.

First take 4 of the apples and give 1 to each child. That leaves 3 more apples to be distributed among the 4 children, now with no restrictions.

So we are left with two problems that are very similar -- dividing 3 apples among 4 children with no restrictions, and dividing 6 oranges among 4 children with no restrictions.

Mathematically, these problems involve finding the number of ways of adding four whole numbers to get particular whole number totals.

A large number of different kinds of problems involve that kind of calculation. A common method for solving such problems is popularly known as "stars and bars". I will demonstrate the process by solving the two cases in your problem.

For the apples, we have 3 apples to be divided among 4 children. We represent the 3 apples with "stars":

***

To model dividing the 3 apples among 4 children, we insert 3 "bars" to separate the stars into 4 groups. For example,

*||*|*

would represent giving 1 more apple each to the 1st, 3rd, and 4th children;

||***|

would represent giving all 3 remaining apples to the 3rd child.

Each different placement of the separating "bars" represents a unique way of distributing the remaining 3 apples to the 4 children.

So the number of ways of distributing the remaining 3 apples to 4 children is the number of ways of arranging the symbols "***|||". By a well-known counting principle, that number of ways is

6C3+=+6%21%2F%28%283%21%29%283%21%29%29+=+20

So there are 20 ways to distribute the remaining 3 apples among the 4 children; and that means there are 20 ways to distribute the 7 apples among 4 children with each child getting at least one.

For the oranges, we use the same process; but now we have 6 oranges to divide among the 4 children.

So for this part of the problem we have 6 stars and again 3 bars; the number of ways of dividing the 6 oranges among the 4 children is

C%289%2C3%29+=+9%21%2F%28%286%21%29%283%21%29%29+=+84

So there are 84 ways to divide the 6 oranges among the 4 children.

And, finally, since distributing the apples and oranges are independent tasks, the total number of ways of distributing 7 apples and 6 oranges to 4 children, with each child getting at least 1 apple, is

20*84 = 1680