In how many ways can $$20$$ oranges be given to four children if each child should get at least one orange?

Question

Infinity Learn · Accepted Answer

The maximum number of oranges that a child can get $$=$$ $$20$$ – $$3$$ $$=$$ $$17$$.Thus, the problem is equivalent to finding the number of integral solutions to the $$equationx1+x2+x3+x4=20where$$ $$1$$≤$$x1$$,$$x2$$,$$x3$$,$$x4$$≤$$17$$, and $$x1$$, $$x2$$, $$x3$$, $$x4$$ denote the number of oranges given to the four children.Hence, the required number of ways $$is=$$ coefficient of $$x20$$ in $$x+x2+x3+$$…$$+x174=$$ coefficient of $$x16$$ in $$1+x+x2+$$…$$+x164=$$ coefficient of $$x16$$ in $$1$$−$$x174(1$$−$$x)$$−$$4=$$ coefficient of $$x16$$ in (1$$−$$x)−$$4=16+4$$−$$1C16=19C3=969$$

In how many ways can 20 oranges be given to four children if each child should get at least one orange?

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