# total number of 4 digit numbers such that the sum of the digits is odd,is (a)2720 (b)5040 (c)3680 (d)4500

148 Points
14 years ago

Dear shaurya sinha

sum of digit will be odd if there is only one odd number or 3 odd number

we have 1,3,5,7,9 odd digit and 0,2,4,6,8 even digit

Case 1    ,only one odd number

a )  odd digit at first place can be filled by 5 ways

and  rest 3 position can be filled by 5*5*5 ways

b  )  odd digit other than 1st palce can be filled by 3C1 *5

and first position can be filled by 4 ways

and rest two position can be filled by 5*5 ways

total ways  =5*5*5*5 +3*5*4*5*5

=2125 ways

case 2 ,exactly 3 odd number

a) even digit at first palec can be filled by 4 ways

and  rest 3 position can be filled by 5*5*5 ways

b  )  even digit other than 1st palce can be filled by 3C1 *5

and  rest 3 position can be filled by 5*5*5 ways

Total number of ways =4*5*5*5 + 3*5*5*5*5

=2374 ways

so total ways =2125+2374 =4500

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