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Interesting Concept

...about "distributions" in Level II session:

In how many ways can 10 cookies be distributed among 5 children such that each child gets at least one cookie.

If cookies are shown as o and children are shown as dividers (markers |), the following diagram is an illustration of one such distribution:

    o o o o o o o o o o
     - - - - - - - - -
      |    |    |    |

In the above illustration '_' shows 9 slots (10 cookies − 1) available for 4 markers (5 children − 1). Only 4 dividers are shown as they divide the cookies into 5 portions.

Based on above, the number of ways to distribute cookies are ⁹C₄.

A special case of this would be, what if there is no restriction that a child gets at least one cookie.

In this case, consider each child to be already having a cookie. Therefore, the problem changes to: in how many ways can 10 + 5 = 15 cookies be distributed among 5 children such that each child gets at least one cookie.

Based on the explanation in the beginning, the number of distributions will be ¹⁴C₄.

Applicable to WK1 #39, WK3 #15

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