If all the numbers which are greater than 600 | vedclass

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(A) The numbers are $4$-digit numbers formed using digits $\{3, 5, 6, 7, 8\}$.Since the numbers are between $6000$ and $10000$,the first digit must be

Vedclass · Accepted Answer

${ }^4 P_3$

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(A) The numbers are $4$-digit numbers formed using digits $\{3, 5, 6, 7, 8\}$.Since the numbers are between $6000$ and $10000$,the first digit must be $6, 7,$ or $8$.Total $4$-digit numbers $= 3 	imes 4 	imes 3 	imes 2 = 72$.For an even number,the last digit must be $6$ or $8$.Case $1$: First digit is $6$. The last digit must be $8$. Remaining $2$ positions can be filled by $3$ digits in $^3 P_2 = 6$ ways.Case $2$: First digit is $7$. The last digit can be $6$ or $8$ ($2$ ways). Remaining $2$ positions can be filled by $3$ digits in $^3 P_2 = 6$ ways. Total $= 2 	imes 6 = 12$.Case $3$: First digit is $8$. The last digit must be $6$. Remaining $2$ positions can be filled by $3$ digits in $^3 P_2 = 6$ ways.Total even numbers $= 6 + 12 + 6 = 24$.Total odd numbers $= 72 - 24 = 48$.Difference $= 48 - 24 = 24$.Since ${ }^4 P_3 = 4 	imes 3 	imes 2 = 24$,the difference is ${ }^4 P_3$.

If all the numbers which are greater than $6000$ and less than $10000$ are formed with the digits $3, 5, 6, 7, 8$ without repetition of the digits,then the difference between the number of odd numbers and the number of even numbers among them is

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