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Question

No. of ways of dividing 80 cards into 5 equal groups of 16 each is


A
80!(16!)5
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B
80!(5!)16
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C
80!(5!)5
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D
80!(16!)5.5!
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Solution

The correct option is D $$\dfrac { 80! }{ (16!)^{ 5 }.5! }$$
We can answer this problem in the following way
First choose 16 cards out of 80, next 16 cards out of (80-16), next 16 cards out of (80-32), and so on till 16 out of 16.
$$\therefore$$ total number of ways of doing this $$= ^{80}C_{16} \times ^{64}C_{16} \times ^{48}C_{16} \times ^{32}C_{16} \times ^{16}C_{16} $$$$= \dfrac{80!}{16!^5}$$
But all the groups are equal so the permutations among them should not be included
$$\therefore$$ The answer is $$=\dfrac{80!}{(16!)^5 \times 5!}$$

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