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  • In how many ways a pack of 52 cards can be divided into four group of 13 cards each?Option: 1 <img alt="\frac{52!}{(13!)^44!}" src="https://learn.car

In how many ways a pack of 52 cards can be divided into four group of 13 cards each?

Option: 1

\frac{52!}{(13!)^44!}


Option: 2

\frac{52!}{(13!)^4}


Option: 3

\frac{52!}{(13!)^24!}


Option: 4

None of these


Answers (1)

best_answer

 

 

DISTRIBUTION OF DISTINCT OBJECTS INTO IDENTICAL PLACES -

To understand this we continue with our previous example with little change, suppose we want to distribute 5 distinct hats in 3 Identical boxes such that each box receives at least 1 hat, this question is exactly what our concept is about, so by solving this we can understand our concept. So to do that we can distribute 3 hats to 1-1 each to all 3 groups, after that we can place remaining 2 hats in one group or 1-1 to 2 groups

So we will have to situation 

1st = 1 1 3, 2nd = 1 2 2 but remember groups are identical

Now these cases are similar to the division of groups where group sizes are given

So, according to 1st case, we can be distributed hats in \\\mathrm{\frac{5!}{(1!)^23! }\cdot\frac{1}{2!} \;ways}

Similarly for 2nd case,\\\mathrm{\frac{5!}{(2!)^21! }\cdot\frac{1}{2!}\; ways}

So the total number of ways of distribution \\\mathrm{=\frac{5!}{(2!)^21! }\frac{1}{2!}\;+ \frac{5!}{(1!)^23!}\times\frac{1}{2!}\; ways}

-

Each group will get 13 cards. Now, the first group can be given 13 cards in ^{52}C_{13} ways.

the second  group can be given 13 cards in ^{39}C_{13} ways (52 - 13 = 39 cards remaining) 

the third  group can be given 13 cards in ^{26}C_{13} ways (39 - 13 = 26 cards remaining) 

the fourth  group can be given 13 cards in ^{13}C_{13} ways (26 - 13 = 39 cards remaining) 

toral number of ways = ^{52}C_{13}\times ^{39}C_{13} \times^{26}C_{13}\times ^{13}C_{13}\times \frac{1}{4!}=\frac{52!}{(13!)^44!}\\

(4 identical groups can be interchanged in 4! ways)

Posted by

Pankaj Sanodiya

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