Q. 395.0( 3 Votes )

# How many natural numbers less than 1000 can be formed from the digits 0, 1, 2, 3, 4, 5 when a digit may be repeated any number of times?

We have to find the possible number of numbers that are formed with the numbers 0, 1, 2, 3, 4, 5 which are less than 1000 when repetition of digits is allowed.

We will use the concept of multiplication because there are sub jobs dependent on each other and are performed one after the other.

First we will make three-digit numbers, There are five choices for hundred’s position because there is zero also which cannot be used in hundred’s place because then our number will become a two digit number instead of a three digit number, there are six choices for the ten’s place because there are a total of six numbers, hundred's placed in which zero was not included but in ten’s place zero is included and in one’s place there are also six choices because repetition is allowed.

The number of ways in which we can form three digit numbers when repetition of digits is allowed along with given numbers 5 × 6 × 6 = 180

Secondly we will make two-digit numbers, There are five choices for ten’s position because there is zero also which cannot be used in ten’s place because then our number will become a one digit number instead of a two digit number, there are six choices for the one’s place because there are a total of six numbers, ten place in which zero was not included but in one’s place zero is included.

The number of ways in which we can form two digit numbers when repetition of digits is allowed along with given numbers 5 × 6 = 30

Thirdly we will form single digit natural numbers, which are given in the question they are five in number, 5 because zero cannot be included because we need natural numbers.

Hence the total number of numbers formed which are less than 1000 and are formed by given numbers are 180+30+5 = 215.

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