Question

How many five- digit numbers can be formed using the digits $0,\,\,1,\,\,2,\,\,3,\,\,4,\,\,5,\,\,6,\,\,7,\,\,8,\,\,9$ if repetitions of digits are not allowed?

Answer 1

Hint: Here we have to form a five- digit number using the digits $0,\,\,1,\,\,2,\,\,3,\,\,4,\,\,5,\,\,6,\,\,7,\,\,8,\,\,9$ and repetitions are not allowed. So, we will form the number by filling the first digit of the five- digit numbers in $9$ ways and the rest of the four digits in $10$ ways. So, we will use the concept of permutation and combination to solve the problem.

Complete step by step answer:
Here, we have to form a five- digit number using the digits $0,\,\,1,\,\,2,\,\,3,\,\,4,\,\,5,\,\,6,\,\,7,\,\,8,\,\,9$ and repetitions are not allowed. The first digit of a number can be filled in $9$ ways except $0$ as we want the number to be five- digits we cannot fill the first digit with $0$. So, the first digit can be filled with $1,\,\,2,\,\,3,\,\,4,\,\,5,\,\,6,\,\,7,\,\,8,\,\,9$.

The second digit of a number can be filled in $9$ ways as one of the numbers is used in the first digit. So, it can be filled with the number which is not used in the first digit. The third digit of a number can be filled in $8$ ways as two digits have been used. So, there are $8$ ways to fill the third digit.

The fourth digit of a number can be filled in $7$ ways as three digits have been used. So, there are $7$ ways to fill the fourth digit. The fifth digit of a number can be filled in $6$ ways as four digits have been used. So, there are $6$ ways to fill the fifth digit.Therefore, the total number of possible numbers are
$ \Rightarrow 9 \times 9 \times 8 \times 7 \times 6$
On multiplying the numbers. We get,
$ \Rightarrow 27,216$

Hence, $27,216$ five-digits numbers can be formed using the digits $0,\,\,1,\,\,2,\,\,3,\,\,4,\,\,5,\,\,6,\,\,7,\,\,8,\,\,9$ without the repetition of digits.

Note: In these types of questions first we have to check the conditions if the repetitions of digits is allowed then we can use all the given numbers but at first digit we cannot use zero as if we use zero in the first digit in the given question it reduces to a four- digit number and we have to form a five-digit number. In order to solve these types of questions we use the concept of permutation and combination. Permutation and combination can be defined as the methods of counting which helps us to determine the number of different ways of arranging and selecting objects out of a given number of objects without actually listing them. Note that we use permutation when order is important and combination is used when order is not important.