Question

Find the number of 4 digit telephone numbers that can be formed using the digits 1, 2, 3, 4, 5, 6 with at least one digit repeated.

Answer 1

Hint: In this question, we first need to find the total number of 4 digit numbers that can be formed using the given numbers which is given by the formula \[{{n}^{m}}\]. Now, find the number of 4 digit numbers that can be formed with all digits different which can be done using the formula \[{}^{n}{{P}_{r}}=\dfrac{n!}{\left( n-r \right)!}\]. Then subtracting the number of 4 digit numbers with all digits different from the total number of 4 digit numbers gives the result.

Complete step-by-step answer:
Now, the given digits in the question are 1, 2, 3, 4, 5, 6
PERMUTATION:
Each of the different arrangements which can be made by taking some or all of a number of things is called a permutation.
The number of ways of arranging n distinct objects in a row taking r at a time is given by
\[{}^{n}{{P}_{r}}=\dfrac{n!}{\left( n-r \right)!}\]
The number of permutations of n different things taken m at a time, when each thing may be repeated any number of times is
\[{{n}^{m}}\]
Now, let us find the total number of 4 digit numbers that can be formed using the given 6 digits 1, 2, 3, 4, 5, 6
Now, on comparing with the above formula we get,
\[n=6,m=4\]
Now, on substituting in the respective formula we get,
\[\Rightarrow {{6}^{4}}\]
Now, on further simplification we get,
\[\Rightarrow 1296\]
Thus, there are 1296 4 digit numbers that can be formed using the digits 1, 2, 3, 4, 5, 6
Now, let us find the 4 digit number that can be formed using 1, 2, 3, 4, 5, 6 in which all the digits are different

Now, form the permutation formula we have
\[{}^{n}{{P}_{r}}\]
Now, on comparing the given values with the formula we get,
\[n=6,r=4\]
Now, on substituting the respective values in the formula we get,
\[\Rightarrow {}^{6}{{P}_{4}}\]
Now, this can be further written as
\[\Rightarrow \dfrac{6!}{\left( 6-4 \right)!}\]
Now, this can be further written in the simplified form as
\[\Rightarrow 6\times 5\times 4\times 3\]
Now, on further simplification we get,
\[\Rightarrow 360\]
Now, the number of 4 digit numbers formed from 1, 2, 3, 4, 5, 6 with at least one digit repeated is given by
\[\Rightarrow 1296-360\]
Now, on further simplification we get,
\[\Rightarrow 936\]
Hence, there are 936 4 digit numbers formed from 1, 2, 3, 4, 5, 6 with at least one digit repeating.

Note: Instead of finding the total numbers that can be formed and then subtracting the number that have no repeated digit we can also solve by considering the cases like 1 digit repeating, 2 digits repeating, 3 digits repeating and all the digits same then add all these to get the result.It is important to note that while finding total numbers possible we need to consider 4 digit numbers in which each digit has 6 ways possible which altogether on multiplication gives the value. Because if we consider 6 in place of 4 and vice versa then the result will be completely incorrect.