What is the number of ways of choosing 4 card | vedclass

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Question

(A) The number of ways to choose $4$ cards from $52$ cards is given by the combination formula $^{52}C_{4}$.$^{52}C_{4} = \frac{52!}{4!48!} = \frac{52

Vedclass · Accepted Answer

$270725$ and $13^{4}$

Vedclass · Answer

(A) The number of ways to choose $4$ cards from $52$ cards is given by the combination formula $^{52}C_{4}$.$^{52}C_{4} = \frac{52!}{4!48!} = \frac{52 	imes 51 	imes 50 	imes 49}{4 	imes 3 	imes 2 	imes 1} = 270725$.To choose $4$ cards such that each belongs to a different suit,we must select $1$ card from each of the $4$ suits (diamonds,hearts,clubs,spades).There are $13$ cards in each suit. The number of ways to choose $1$ card from each suit is $^{13}C_{1} 	imes ^{13}C_{1} 	imes ^{13}C_{1} 	imes ^{13}C_{1} = 13 	imes 13 	imes 13 	imes 13 = 13^{4} = 28561$.Thus,the total ways are $270725$ and the ways to choose $4$ cards from different suits are $13^{4}$.

What is the number of ways of choosing $4$ cards from a pack of $52$ playing cards? In how many of these do the four cards belong to four different suits?

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