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Lots across desks

CombinatoricsEasy

A market maker splits $7$ identical lots among $3$ distinct desks. A desk may receive any number of lots, including $0$ . How many different allocations are there?

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Hint

Lots are identical and desks are distinct, so this is a stars and bars count.

Solution

Count the solutions of $x 1 + x 2 + x 3 = 7$ with $x i \geq 0$ .
Stars and bars gives $(2 7 + 2)$ allocations.
This equals $(2 9) = 36$ .
The number of allocations is $36$ .

Answer: $36$

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