Latest UPSC Aptitude Question SOLUTION: 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.

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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.

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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.


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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

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

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

99
Ok

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