1.
Find the number of times 5 will be written while listing integers from 1 to 1000.
a) 270
b) 290
c) 280
d) 300
Answers
5 will be written 300 times while listing integers from 1 to 1000.
Step-by-step explanation:
Firstly, we will find the numbers having at least one digit as 5 in the range [1, 1000]
We can do it by finding the numbers which do not contain ‘5’ as a digit in any place.
As we know, there are 9 options in [0–9] excluding 5 for the unit, tenth & hundreds place.
Total numbers not having 5 = 9 * 9 * 9 = 729
Thus, the count of numbers having ‘5’ as a digit in any place = 1000 - 729
= 271
As we know, 271 is the count of numbers having at least ‘5’ as one digit, and not the complete count of digit ‘5’ in the range [1–1000].
These 271 numbers will also include those numbers having ‘5’ twice or thrice.
Numbers having ‘5’ exactly twice = 55, 155, 255, 355, 455, 505, 515, 525, 535, 545, 550, 551, 552, 553, 554, 556, 557, 558, 559, 565, 575, 585, 595, 655, 755, 855, 955 (one ‘5’ is already counted above so we will count other one ‘5’)
Numbers having ‘5’ exactly thrice = 555 (one ‘5’ is already counted, so we will count the other 2 fives)
Thus, the number of digit ‘5’ in (1–1000) = 271 + 27(one in the numbers having them twice) + 2(in the number having 5 thrice) = 300.
Learn more: Integers
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