How many odd numbers, greater than 600000 can be formed from the digits 5,6,7,8,9,0 if(a) Repetitions are allowed,(b) Repetitions are not allowed.

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Filo · Accepted Answer

Repetition allowed. 5,6,7,8,9,0 : Six digits.The first place can be filled by 6,7,8,9 i.e. in 4 ways as the number is to be greater than 600000. The last place can be filled in by 5, 7, 9 i.e. 3 ways as the number is to be odd and because of repetition. Hence the number of ways of filling the 1st and the last place is4×3=12 waysWe have to fill in the remaining 4 places of the six digit number i.e. 2nd, 3rd, 4th and 5th place. Since repetition is allowed each place can be filled in 6 ways. Hence the 4 places can be filled in 6×6× 6×6=1296 ways.Hence by fundamental theorem the total numbers will be1296× 12=15552.Repetition not allowed.1st PlaceLast (Number to be odd)65,7,9=3 ways85,7,9= 3 ways75,9=2 ways (not 7)95,7=2 ways (not 9)Total number of ways of filling the first and the last place under given condition is3+3+2+2=10 ways.(It was 12 when repetition was allowed).Having filled in the first and the last places in 10 ways, we can fill in the remaining 4 places out of 4 digits (repetition not allowed) in 4! = 24 ways.Total number of ways is 10×24=240

1st Place	Last (Number to be odd)
$6$	$5, 7, 9 = 3$ ways
$8$	$5, 7, 9 = 3$ ways
$7$	$5, 9 = 2$ ways (not 7)
$9$	$5, 7 = 2$ ways (not 9)

Question Text	How many odd numbers, greater than $600000$ can be formed from the digits $5, 6, 7, 8, 9, 0$ if (a) Repetitions are allowed, (b) Repetitions are not allowed.
Answer Type	Text solution:1
Upvotes	150

How many odd numbers, greater than $600000$ can be formed from the digits $5, 6, 7, 8, 9, 0$ if
(a) Repetitions are allowed,
(b) Repetitions are not allowed.

Text solutionVerified

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