How many even numbers of four digits can be f | vedclass

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(C) To form a four-digit even number,the units place must be filled by $0, 2, 4,$ or $6$.Case $I$: When $0$ is at the units place.The units place can

Vedclass · Accepted Answer

$420$

Vedclass · Answer

(C) To form a four-digit even number,the units place must be filled by $0, 2, 4,$ or $6$.Case $I$: When $0$ is at the units place.The units place can be filled in $1$ way. The remaining $3$ places can be filled using the remaining $6$ digits $(1, 2, 3, 4, 5, 6)$ in $^6P_3$ ways.Number of ways $= 1 	imes ^6P_3 = \frac{6!}{3!} = 6 	imes 5 	imes 4 = 120$.Case $II$: When $0$ is not at the units place.The units place can be filled by $2, 4,$ or $6$ in $3$ ways. The thousands place cannot be $0$ and cannot be the digit used in the units place,so it can be filled in $5$ ways (out of $6$ remaining digits). The remaining $2$ places can be filled by the remaining $5$ digits in $^5P_2$ ways.Number of ways $= 3 	imes 5 	imes ^5P_2 = 15 	imes \frac{5!}{3!} = 15 	imes 20 = 300$.Total even numbers $= 120 + 300 = 420$.

How many even numbers of four digits can be formed with the digits $0, 1, 2, 3, 4, 5$ and $6$,if no digit is used more than once?

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