# How many different ways can she leave these paintings to her three heirs?

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• Jan 7th 2009, 06:25 PM
Yan
How many different ways can she leave these paintings to her three heirs?
An art collector, who owns 10 paintings by famous artists, is preparing her will. In how many different ways can she leave these paintings to her three heirs?

I had work on this problem for a long time but the answer still not match with on that shows on the solution manual. and the correct answer is 59,049. Can you show me how to do it?
• Jan 7th 2009, 08:37 PM
mr fantastic
Quote:

Originally Posted by Yan
An art collector, who owns 10 paintings by famous artists, is preparing her will. In how many different ways can she leave these paintings to her three heirs?

I had work on this problem for a long time but the answer still not match with on that shows on the solution manual. and the correct answer is 59,049. Can you show me how to do it?

Are you familiar with Stirling numbers?: http://www.cargalmathbooks.com/17%20...%20type%20.pdf

The answer to your question is $3! \cdot {10 \choose 3} + 3 \cdot 2! \cdot {10 \choose 2} + 3 \cdot {10 \choose 1}$

where ${n \choose k}$ are Stirling numbers.
• Jan 7th 2009, 10:46 PM
Soroban
Hello, Yan!

Quote:

An art collector, who owns 10 paintings by famous artists, is preparing her will.
In how many different ways can she leave these paintings to her three heirs?

For each of the ten paintings, there are three choices of heirs.

Hence, there ar: . $3^{10} \,=\,59,\!049$ possible ways to distribute the paintings.