Question

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; 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

Answer 1

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

Answer 2

Answer:

case 1- 300-400

304,314,324,334,341,..343,345,..349,354,364,374,384,394,400

case 2- 401-500

401-410- 9 numbers

Step-by-step explanation:

so there are 19 numbers in first case

in second case: excluding 440-449 the total numbers from 401-490 will be 9*8=72

and from 491 - 500 there are 8

so now 19+72+8=99