total number of 4 digit numbers such that the sum of the digits is odd,is

• (a)2720
• (b)5040
• (c)3680
• (d)4500

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