Find the number of 4-digit numbers that can b | vedclass

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Question

(A) The total number of $4$-digit numbers formed using the digits $1, 2, 3, 4, 5$ without repetition is given by the number of permutations of $5$ dis

Vedclass · Accepted Answer

$120, 48$

Vedclass · Answer

(A) The total number of $4$-digit numbers formed using the digits $1, 2, 3, 4, 5$ without repetition is given by the number of permutations of $5$ distinct digits taken $4$ at a time.This is calculated as $^5P_4 = \frac{5!}{(5-4)!} = \frac{120}{1} = 120$.For a number to be even,the units place must be filled with an even digit,which in this set is either $2$ or $4$.There are $2$ choices for the units place.Once the units place is filled,the remaining $3$ places must be filled using the remaining $4$ digits.The number of ways to fill these $3$ places is $^4P_3 = \frac{4!}{(4-3)!} = 4 	imes 3 	imes 2 = 24$.By the multiplication principle,the total number of even $4$-digit numbers is $2 	imes 24 = 48$.

Find the number of $4$-digit numbers that can be formed using the digits $1, 2, 3, 4, 5$ if no digit is repeated. How many of these will be even?

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