how many numbers are possible between 100 to 999 where middle digit is smaller than other two. e.g. 102?

• 10__=1*1*9=9
  20__=1*1*9(when middle digit is 0)+1*1*8(when middle term is 1)=9+8
  30__=1*1*9(when middle digit is 0)+1*1*8(when middle term is 1)+1*1*7(when middle term is 2)=9+8+7
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  9__=9+8+7+6+5+4+3+2+1
  
  =>9*9+8*8+7*7+6*6+5*5+4*4+3*3+2*2+1*1=285 Ans
• 7 years agoHelpfull: Yes(29) No(1)
• in case repetition is of allow
  10_ last place can be 1,2,3,4,5,6,7,8,9= total 9
  2_ in second place we can place either 0,1 when place 0 there 9 option when place 1 so 2,3,4,5,6,7,8,9 there is 8 option = total 9+7
  3 _ second place can be 0,1,2 and similar to above total= 9+8+7
  ;
  ;
  9_ second place 0,1,2,3,4,5,6,7,8 total=9+8+7+6+5+4+3+2+1
  so 9 count 9 time , 8 count 8 time and so on
  sum=9*9+8*8+7*7+......1*1
  this is in A.p so sum=n(n+1)(2n+1)/6
  sum=9(9+1)(2*9+1)/6=285
  
  in case repetition is not allowed
  sum=8(8+1)(2*8+1)/6=204
• 7 years agoHelpfull: Yes(6) No(0)
• from 201 to 300 it will be 9 only
  then again from 301 to 400 it will be 9 and 8
  ....
  901 to 999 it will be 9 and 8 times 8
  
  after calculating these total will be
  369
• 7 years agoHelpfull: Yes(3) No(2)
• 285
  add square from 1 to 9
• 6 years agoHelpfull: Yes(3) No(1)
• 10_ =9
  2__ =9+8
  3__=9+8+7
  4__=9+8+7+6
  ;
  ;
  ;
  9__=9+8+7+6+5+4+3+2+1
  them sum all these=9+17+24+30+35+39+42+44+45=285
  ans=285
• 7 years agoHelpfull: Yes(2) No(5)
• 1. If middle place is 0 than 1st and third place can be filled by nos 1-9 => total = 9*9
  2. Similarly if middle=1 then 1st and 3rd place can be filled by nos 2-8 => total = 8*8
  3. Similarly middle=2, total = 7*7
  4. Similarly middle=3, total = 6*6
  Similarly middle=4, total = 5*5
  Similarly middle=5, total = 4*4
  Similarly middle=6, total = 3*3
  Similarly middle=7, total = 2*2
  Similarly middle=8, total = 1*1
  Note: We can not have 9 digit at middle position as per the condition given in question.
  So the series would be = 1*1+2*2+3*3+4*4+5*5+6*6+7*7+8*8+9*9
  For this series we can apply formula = n(n+1)(2n+1) / 6, therefore after putting the values and solving them answer would be = 285.
• 6 years agoHelpfull: Yes(1) No(0)