How many combinations can there be

Question

There are 15 distinct A stamps and 15 distinct B stamps for a total of 30 distinct stamps. The album is put together using a combination of nine A or B stamps and six of the other for a total of 15. How many combinations of the album would there be if you used all of the A & B stamps with the nine / six stamp split.

This is an actual situation in which a stamp album that was printed in 1915 and thought to have only 15 stamps has been discovered to have a second set that was mixed into the album with a 9 stamp / six stamp split from both sets.

Any help would be greatly appreciated!

What do you mean "used all of the A&B stamps"? The album has how many stamps? — Kal S., May 02 '14 at 05:05

score 2 · Accepted Answer · answered May 02 '14 at 05:39

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The number of ways to pick $k$ stamps from $A$ is $\binom{15}k$. Of course, the same holds for $B$. Thus, the number of ways to select $9$ stamps from $A$ and $6$ stamps from $B$ is $$N=\binom{15}9\binom{15}6=\binom{15}6^2.$$ To account for the other possibility, namely choosing $6$ from $A$ and $9$ from $B$, we simply have to double $N$. The answer is $2\binom{15}6^2$ a number which I would prefer not to write in decimal notation. :)

answered May 02 '14 at 05:39

lol wut

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$$50,100,050 $$ – DanielV May 02 '14 at 06:07
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@DanielV Has no qualms though. It looked bigger on my hand calculator tbh. – lol wut May 02 '14 at 06:14
I used wxMaxima, I thought it would be bigger too, as factorials tend to explode. You did all the hard work though; especially interpreting the question. – DanielV May 02 '14 at 06:15
Thanks so much for your answer. It is hard to believe there would be so many combinations of this set out there. That is amazing! – Robert May 03 '14 at 03:36

How many combinations can there be

1 Answers1