We have 8 pieces of strawberry candy and 7 pieces of pineapple candy. In how many ways can we distribute this candy to 4 kids?

Mellie
Apr 30, 2015

#4**+6 **

I don't know Mellie, but

If there are 8 strawberry candies and each child gets at least one of them then that is (8+4-1)C8 = 11C8 = 165 ways

If there are 7 pineabpple candies and each child gets at least one of them then that is (7+4-1)C7 = 10C7=120

So if every child gets at least on of each then that would be 11C8*10C7 = 165*120 = **19800 ways**

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NOW what about if one or more children get no pinapple lollies or no strawberry lollies or no lollies at all :/

If there are 8 strawberry candies and each child gets at least one of them then that is (8+4-1)C8 = 11C8 = 165 ways

If one child gets 0 strawbs then this can be done in (8+3-1)C8 = 10C8=45 ways

If two children gets 0 strawbs then this can be done in (8+2-1)C8 = 9C8=9 ways

If three children get 0 strawbs then this can be done in (8+1-1)C8=1 way (which makes sense because one child is getting all the strawb lollies.

So the number of ways that the strawberry lollies can be distributed is 165+45+9+1 = 220 ways

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If there are 7 pineapple candies and each child gets at least one of them then that is (7+4-1)C7 = 10C7=120

If one child gets 0 pineapple candies then this can be done in (7+3-1)C7 = 9C7= 36 ways

If two children gets 0 pineapple candies then this can be done in (7+2-1)C7 = 8C7=8 ways

If three children get 0 pineapple candies then this can be done in 1 way

So the number of ways that the pineapple candies can be distributed is 120+36+8+1 = 165 ways

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so the number of ways the lollies can be distributed is **220*165=36,300 ways **

(some children might get none)

Melody
May 6, 2015

#5**+5 **

.

$$\ ${Use this formula for each set then multiply the sets}\\

\displaystyle \left( {\begin{array}{*{20}c} N + K - 1 \\ K-1 \\ \end{array}} \right)\; =\; distributions \\

\displaystyle \left( {\begin{array}{*{20}c} 8+4-1\\ 4-1 \\ \end{array}} \right)\; =\; 165 \\

\displaystyle \left( {\begin{array}{*{20}c} 7+4-1\\ 4-1 \\ \end{array}} \right)\; =\; 120 \\

\ 165 * 120\; =\; 19,800$$

Nauseated
May 6, 2015