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

Hint: 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.