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Question

In how many ways can a pack of $$52$$ cards be divided in $$4$$ sets, three of them having $$17$$ cards each and fourth just $$1$$ card?


A
52!3!(17!)3
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B
52!(17!)3
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C
52!(17!)4
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D
52!3!(17!)
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Solution

The correct option is A $$\displaystyle\frac{52!}{3!(17!)^3}$$
First we divide $$52$$ cards into two sets which contains $$1$$ and $$51$$ cards respectively is 
$$\quad\displaystyle\frac{52!}{1!\space 51!}$$
Now
$$51$$ cards can be divided equally in three sets each contains $$17$$
cards (Here order of sets is not important) in
$$\displaystyle\frac{52!}{3!(17!)^3}$$ ways
Hence, the required number of ways 
$$\quad = \space \displaystyle\frac{52!}{1! \space 51!} \times \displaystyle\frac{51!}{3!\space(17!)^3}$$

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