mathmath333
  • mathmath333
A library has 20 copies of book A ,12 copies each of book B part 1 and part 2 , 5 copies of book C part 1 ,part 2 and part 3 and single copy of book D, book E, book F. In how many ways can these book be distributed
Mathematics
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SOLVED
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katieb
  • katieb
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mathmath333
  • mathmath333
\(\large \color{black}{\begin{align} & \normalsize \text{ A library has 20 copies of book A ,12 copies each of}\hspace{.33em}\\~\\ & \normalsize \text{ book B part 1 and part 2 , 5 copies of book C part 1 }\hspace{.33em}\\~\\ & \normalsize \text{ ,part 2 and part 3 and single copy of book D, book E,}\hspace{.33em}\\~\\ & \normalsize \text{ book F. In how many ways can these book be distributed}\hspace{.33em}\\~\\ & a.)\ \dfrac{62!}{20!12!5!} \hspace{.33em}\\~\\ & b.)\ 62! \hspace{.33em}\\~\\ & c.)\ \dfrac{62!}{(37)^{3}} \hspace{.33em}\\~\\ & d.)\ \dfrac{62!}{20!(12!)^{2}(5!)^{3}} \hspace{.33em}\\~\\ \end{align}}\)
Empty
  • Empty
Are parts considered books
ParthKohli
  • ParthKohli
Is it D?

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mathmath333
  • mathmath333
Yes answer is given \(D.)\)
ParthKohli
  • ParthKohli
OK. It's because it is D.
mathmath333
  • mathmath333
Example \(\text{ Book C part 1}\) and \(\text{ Book C part 2}\) are two different books
Empty
  • Empty
\[\frac{(1*20+2*12+3*5+1*1+1*1+1*1)!}{20! 12!^25!^3}\]
ParthKohli
  • ParthKohli
yeah, really needs no explanation.
mathmath333
  • mathmath333
i didnt understand the denominator \(\dfrac{(1*20+2*12+3*5+1*1+1*1+1*1)!}{\color{red}{20! 12!^25!^3}}\)
Empty
  • Empty
There are 20! ways to arrange the A books. But since these are indistinguishable, we divide this quantity out of the total since 62! treats each book as if they were independent when they're not.
ParthKohli
  • ParthKohli
If I have these letters and I want to find the number of arrangements... ``` AAABBC ``` I will first find the number of arrangements of five distinct letters (5!) then divide by 3!*2! because A repeats thrice and B repeats twice.
ParthKohli
  • ParthKohli
Sorry, that's six distinct letters.
Empty
  • Empty
It's probably better to think through this question with a small size, well @ParthKohli is on it and I'm not in India so it's 4AM here later lol.

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