The total number of 5 digit numbers of different digits in which the middle digit is the largest is

Question

Professor PC Thomas Classes · Answer

5 digit numbers containing 0 . Select 5 digits including 0 . This can be done in ${ }^9 \mathrm{C}_4$ ways. Write the greatest digit in the middle. The remaining 4 digits which include 0 can be arranged in $3 	imes 3!=18$ ways. Total ${ }^9 \mathrm{C}_4 	imes 18$.Numbers not containing 0.5 non zero digits can be chosen in ${ }^9 \mathrm{c}_5$ ways. Write the greatest in the middle place. The remaining 4 digits can be arranged in $4!=$ 24 ways. Total ${ }^9 \mathrm{C}_5 	imes 24$. ∴ Total number of 5 digit numbers with the greatest in the middle place is \ $ { }^9 \mathrm{C}_4 	imes 18+{ }^9 \mathrm{C}_5 	imes 24=\left({ }^9 \mathrm{C}_4ight) 	imes 42 $