GATE Overflow Test Series | Discrete Mathematics | Test 2 | Question: 12

Question

A game of cards is played among $4$ friends. After the initial split of $52$ cards, the tallest person told that he has got $4$ Aces and all the remaining $9$ cards are of the same suit. Total number of ways in which this is possible is $\_\_\_$

Answer 1

$4$ Aces out of $4$ can be chosen only in $1$ way.

There are $4$ possible suits in a deck of cards and we can choose one in $4$ ways.

A suit has $13$ cards and excluding ACE, $12.$ We can select $9$ cards from them in ${}^{12}C_9$ ways.

So, total number of ways $ = 1 \times 4 \times {}^{12}C_9 $

$\qquad =4 \times {}^{12}C_3$

$\qquad = 4 \times \frac{12\times 11\times 10}{6} = 880.$

Comments

here we have consider that out of 4 friends who get all Ace so first 4c1 ,then selecting a suit will be 4c1 and then for all Ace  we have  only 1  choice and all suits now left with only 12 cards and out of this 12 cards they belongs to same suit will be 12c9 . so finally

4C1 * 4C1 * 1* 12C9 = 3520

please someone check and clear my doubt!!!

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I see that the confusion here is regarding the choice of the “one person”. If he can be chosen answer is $880\times 4=3520$ and if he’s fixed answer is $880.$

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I have made the statement clearer now as “one person” was a bit ambiguous.