Q. 1.13 - A First Course in Probability

Question

An art collection on auction consisted of $4$ Dalis, $5$ van Goghs, and $6$ Picassos. At the auction were 5 art collectors. If a reporter noted only the number of Dalis, van Goghs, and Picassos acquired by each collector, how many different results could have been recorded if all of the works were sold?

Step-by-Step Solution

Answer

No. of recorded results if all of the works were sold is $1,852,200$.

Step 1. Find the no. of results in case of Dalis.

Let the no. of works be $X_1, X_2, X_3, X_4, X_5$

In case of Dalis $X_1+X_2+X_3+X_4+X_5=4$

No. of solutions $=\left(\begin{array}{c}4+5-1\\ 4\end{array}\right)=\left(\begin{array}{c}8\\ 4\end{array}\right)=70$

Step 2. Find the no. of results in case of Van Goghs.

Let the no. of works be $X_1, X_2, X_3, X_4, X_5$

In case of Van Goghs $X_1+X_2+X_3+X_4+X_5=5$

No. of solutions $=\left(\begin{array}{c}5+5-1\\ 4\end{array}\right)=\left(\begin{array}{c}9\\ 4\end{array}\right)=126$

Step 3. Find the no. of results in case of Picassos.

Let the no. of works be $X_1, X_2, X_3, X_4, X_5$

In case of Picassos $X_1+X_2+X_3+X_4+X_5=6$

No. of solutions $=\left(\begin{array}{c}6+5-1\\ 4\end{array}\right)=\left(\begin{array}{c}10\\ 4\end{array}\right)=210$

Step 4. Find the no. of recorded results if all of the works were sold.

Therefore, the no. of recorded results if all of the works were sold $=70\times 126\times 210=1,852,200$.