Question

Determine the total number of natural numbers less than 7,000 which can be formed by using the digits \left\\{ 0,1,3,7,9 \right\\} where repetition of digits is allowed.
(a) 250
(b) 374
(c) 372
(d) 375

Answer 1

In this question, we have to determine number of natural numbers less than 7,000 which can be formed by using the digits \left\\{ 0,1,3,7,9 \right\\} where repetition of digits allowed. For that we have to first find the total number of 1 digit numbers that can be formed using \left\\{ 0,1,3,7,9 \right\\} which will be equal to 4. Then we will find the total number of 2 digit numbers that can be formed using \left\\{ 0,1,3,7,9 \right\\}. We have to take care of the fact that repetition of digits is allowed and also while forming a two or more digit number, the number cannot start with 0.
We will then find the total number of 3 digit numbers that can be formed using \left\\{ 0,1,3,7,9 \right\\} and then find the total number of 4 digit numbers that can be formed using \left\\{ 0,1,3,7,9 \right\\}.Then we will add all these obtained values to determine number of natural numbers less than 7,000 which can be formed by using the digits \left\\{ 0,1,3,7,9 \right\\}.

Complete step-by-step answer:
Let us suppose that $S$ denotes the set of elements S=\left\\{ 0,1,3,7,9 \right\\} where the number of elements in the set $S$ is 5.
Now we have to determine natural numbers less than 7,000 which can be formed by using the elements of the set S=\left\\{ 0,1,3,7,9 \right\\}.
Now we will first determine the total number of 1 digit numbers that can be formed using \left\\{ 0,1,3,7,9 \right\\}.
Since we know that 0 is not a natural number.
Thus the total number of 1 digit numbers that can be formed using \left\\{ 0,1,3,7,9 \right\\} is given by
$5-1=4$
Then we will determine the total number of 2 digit numbers that can be formed using \left\\{ 0,1,3,7,9 \right\\}.
Since we are given that repetition of digits is allowed and also we know that ,while forming a two or more digit number, the number cannot start with 0.
Thus in order to make a two digit number using the elements in the set S=\left\\{ 0,1,3,7,9 \right\\}, the number of choices for the digit in ${{10}^{th}}$ place of the two digit number is given by
$5-1=4$
And the number of choices for the digit in ones place of the two digit number is given by
$5$
Thus the total number of two digit number using the elements in the set S=\left\\{ 0,1,3,7,9 \right\\} is given by
$4\times 5=20$
Then we will determine the total number of 3 digit numbers that can be formed using \left\\{ 0,1,3,7,9 \right\\}.
Since the 3 digit number cannot start with 0, thus the number of choices for the digit in ${{100}^{th}}$ place of the two digit number is given by
$5-1=4$
And the number of choices for the digit in ${{10}^{th}}$ place of the two digit number is given by
$5$
And the number of choices for the digit in ones place of the two digit number is given by
$5$
Thus the total number of three digit number using the elements in the set S=\left\\{ 0,1,3,7,9 \right\\} is given by
$4\times 5\times 5=100$
Finally we will calculate the total number of 4 digit numbers less than 7000 that can be formed using \left\\{ 0,1,3,7,9 \right\\}.
Since the 4 digit number cannot start with0 and also the number cannot start we 7 and 9 otherwise the 4 digit number will be greater than or equal to 7000, thus the number of choices for the digit in ${{1000}^{th}}$ place of the two digit number is given by
$5-3=2$
And the number of choices for the digit in ${{100}^{th}}$ place of the two digit number is given by
$5$
And the number of choices for the digit in ${{10}^{th}}$ place of the two digit number is also given by
$5$
And the number of choices for the digit in ones place of the two digit number is given by
$5$
Thus the total number of four digit number less than 7000 that can be formed using the elements in the set S=\left\\{ 0,1,3,7,9 \right\\} is given by
$2\times 5\times 5\times 5=250$
Therefore the total number of natural numbers number less than 7000 that can be formed using the elements in the set S=\left\\{ 0,1,3,7,9 \right\\} is given by the sum
$4+20+100+250=374$
Hence the total number of natural numbers less than 7,000 which can be formed by using the digits \left\\{ 0,1,3,7,9 \right\\} where repetition of digits allowed is equal to 374.

So, the correct answer is “Option (b)”.

Note: In this problem, in order to determine the total number of natural numbers less than 7,000 which can be formed by using the digits \left\\{ 0,1,3,7,9 \right\\} where repetition of digits, take care of the facts that 0 is not a natural number and a two or more digit number, the number cannot start with 0.