Everyone Loves Cupcakes

Probability Level 4

It's Alice's birthday and she invites 9 of her best friends over for her birthday party. Alice and her friends are all having a great time when Alice's dad brings out a tray of cupcakes. There are 10 cupcakes on the tray, one for each kid at the party. Everyone lines up anxiously awaiting to get a cupcake, and one by one each kid takes a cupcake from the tray.

Unfortunately, some of Alice's friends are greedy! They take more than their fair share of cupcakes, and Alice, who's last in line, doesn't get one! How many ways could this have happened? In other words, how many ways could we arrange 9 numbers such that their sum is 10?

Details and Assumptions

  • Alice's friends could have taken anywhere from 0 cupcakes to all 10 cupcakes.


The answer is 43758.

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Garrett Clarke by Garrett Clarke , 25, USA

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1 solution

Shandy Rianto
Jul 8, 2015

We simply need to find the number of non-negative integral solutions of

x 1 + x 2 + x 3 + x 4 + x 5 + x 6 + x 7 + x 8 + x 9 = 10 x_{1} + x_{2} + x_{3} +x_{4} + x_{5} + x_{6} + x_{7} + x_{8} +x_{9} = 10

which has ( 18 10 ) = 43758 {18 \choose 10} = \boxed{43758} solutions

1 Helpful 0 Interesting 0 Brilliant 0 Confused

Moderator note:

It's better to clarify how you obtain the ( 18 10 ) {18\choose 10} .

Bonus question : Is there an almost similar method to solve this?

Hint : Don't leave out Alice.

That's a part of Combinations with Repetition .

Shandy Rianto - 5 years, 11 months ago

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