Question 93

Find the number of ways of distributing five identical chocolates and six identical cookies among three children so that each child gets at least one cookie.

The number of ways of distributing $$n$$ items among $$r$$ persons is $$^{n+r-1}C_{r-1}$$

If at least one item is given to each person, then the number of ways is $$^{n-1}C_{r-1}$$

As there are five chocolates and three children, the number of ways is $$^{n+r-1}C_{r-1}=^{5+3-1}C_{3-1}=^7C_2=21$$

For cookies, there are six cookies and we need to give at least 1 to each child, the number of ways is $$^{n-1}C_{r-1}=^{6-1}C_{3-1}=^5C_2=10$$

So, the total number of ways is $$21\times10=210$$

Hence, the answer is 210.

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