How many numbers lie between 300 and 500 in which 4 comes only one time? [UPSC 2009(CS-P)]
a) 99
b) 100
c) 110
d) 120

• They are all of the form
  
  "4AB", "34B", and "3B4", where A nor B is 4
  
  Case 1: The number of 3-digit numbers of type "4AB"
  
  We can choose A as any of the 9 digits {0,1,2,3,5,6,7,8,9}
  We can choose B as any of the 9 digits {0,1,2,3,5,6,7,8,9}
  
  So there are 9×9 = 81 ways for Case 1.
  
  ----------------------------------------------------
  
  Case 2: The number of 3-digit numbers of type "34B"
  
  We can choose B as any of the 9 digits {0,1,2,3,5,6,7,8,9}
  
  So there are 9 ways for Case 2.
  
  ------------------------------
  
  Case 3: The number of 3-digit numbers of type "3B4"
  
  Same as Case 2: We can choose B as any of the 9 digits {0,1,2,3,5,6,7,8,9}
  
  So there are 9 ways for Case 2.
  
  
  -----------------------------------------
  
  Total 81+9+9 = 99
  
  Here are all 99 in a 9 by 11 array:
  
  304, 314, 324, 334, 340, 341, 342, 343, 345, 346, 347
  348, 349, 354, 364, 374, 384, 394, 400, 401, 402, 403
  405, 406, 407, 408, 409, 410, 411, 412, 413, 415, 416
  417, 418, 419, 420, 421, 422, 423, 425, 426, 427, 428
  429, 430, 431, 432, 433, 435, 436, 437, 438, 439, 450
  451, 452, 453, 455, 456, 457, 458, 459, 460, 461, 462
  463, 465, 466, 467, 468, 469, 470, 471, 472, 473, 475
  476, 477, 478, 479, 480, 481, 482, 483, 485, 486, 487
  488, 489, 490, 491, 492, 493, 495, 496, 497, 498, 499
• 6 years agoHelpfull: Yes(3) No(0)
• 1st of all 300 to 399,
  3n4 & 34n in which n is filled by 18..........(1)
  now 400 to 499
  4n4 & 44n in which n is filled by 19,
  so out of 100 (400-499) 19 numbers have one more than 4.
  =100-19= 81.......(2)
  ans.=81+18=99
• 9 years agoHelpfull: Yes(1) No(1)
• The number lies between 300 and 500 in which 4 comes only one time are:
  300 – 399 = 18
  400 – 410 = 10
  411 – 420 = 9
  421 – 430 = 9
  431 – 439 = 8
  450 – 460 = 10
  461 – 470 = 9
  471 – 480 = 9
  481 – 490 = 9
  491 – 500 = 8
  ∴ Required number are:
  18 + 10 + 9 + 9 + 8 + 10 + 9 + 9 + 9 + 8 = 99
  
• 8 years agoHelpfull: Yes(0) No(2)
• 99
  Ok
• 4 years agoHelpfull: Yes(0) No(0)
• the answer is option a)99
  from 300 the outcomes of four is,
  1.304,314,324,334,354,364,374,384,394=9
  2.and 340 to 349 the outcomes of 4 is 9
  3.400 to 499(100) except 404,414,424,434,454,464,474,484,494 and 440 to 449 is 19 that is 100-19=81
  4.81+9+9=99
• 4 years agoHelpfull: Yes(0) No(0)