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Choose the correct number of ways in which 15 different books can be divided into five heaps of equal number of books
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Verified Answer
The correct answer is:
$\frac{15!}{5!(3!)^5}$
There are five heaps of an equal number of books.
Thus each heap contains $\frac{15}{5}=3$ books.
So the required number of ways
$={ }^{15} C_3 \times{ }^{12} C_3 \times{ }^9 C_3 \times{ }^6 C_3 \times{ }^3 C_3 \times$ $\times \frac{1}{5!}$
$=\frac{15!}{12!\times 3!} \times \frac{12!}{9!\times 3!} \times \frac{9!}{6!\times 3!} \times \frac{6!}{3!\times 3!} \times \frac{3!}{0!\times 3!}$ $\times \frac{1}{5!}$
$=\frac{5!}{\left[5!(3!)^5\right]}$
Thus each heap contains $\frac{15}{5}=3$ books.
So the required number of ways
$={ }^{15} C_3 \times{ }^{12} C_3 \times{ }^9 C_3 \times{ }^6 C_3 \times{ }^3 C_3 \times$ $\times \frac{1}{5!}$
$=\frac{15!}{12!\times 3!} \times \frac{12!}{9!\times 3!} \times \frac{9!}{6!\times 3!} \times \frac{6!}{3!\times 3!} \times \frac{3!}{0!\times 3!}$ $\times \frac{1}{5!}$
$=\frac{5!}{\left[5!(3!)^5\right]}$
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