Question

How many 4-digit numbers can be formed from digits 1, 1, 2, 2, 3, 3, 4, 4, 5, 5.

Answer 1

In the above question we need to find the number of ways to form $4$-digit numbers. If all four digits are different, we use $^5{P_4}$. If there is one repeated digit, there are $5$ ways to choose which digit is repeated. Then there are $^4{C_2} = 6$ ways to place the repeated digit and $^4{P_2}$ = 12 ways to place the non – repeated digits. And similarly, we arrange numbers when there are two repeated digits.

Complete step by step solution:
According to the question, we have to from $4$-digit number from digits $1, 1, 2, 2, 3, 3, 4, 4, 5, 5$.
We can form $4$-digit number in three ways –
When all the four digits are different, then the number of 4-digit number can be formed is = $^5{P_4}$
$= \dfrac{{5!}}{{1!}}$
$= \dfrac{{5 \times 4 \times 3 \times 2 \times 1}}{1}$
$= 120$ways.
When two digits are same and another two digits are different, then the number of 4-digit numbers can be formed is =$^5{C_1}{ \times ^4}{C_2}{ \times ^4}{P_2}$
$= \dfrac{{5!}}{{4! \times 1!}} \times \dfrac{{4!}}{{2! \times 2!}} \times \dfrac{{4!}}{{2!}}$
$= \dfrac{{5 \times 4!}}{{4! \times 1}} \times \dfrac{{4 \times 3 \times 2!}}{{2! \times 2 \times 1}} \times \dfrac{{4 \times 3 \times 2!}}{{2!}}$
$= 5 \times 6 \times 12$
$= 360$ ways
When there are two repeated digits, then the number of 4-digit numbers can be formed is = $^5{C_2}$
$= \dfrac{{5!}}{{3! \times 2!}}$
$= \dfrac{{5 \times 4 \times 3!}}{{3! \times 2 \times 1}}$
$= 10$
There are $6$ ways to place the first repeated digit and $1$ way to place the second repeated digit.
Thus, we have;
Total number of ways when there are two repeated digits are$= 10 \times 6 \times 1 = 60$ways
Total number of ways to form 4-digit numbers from given digits are$= 120 + 360 + 60$
$= 580$ ways.

$\therefore$Total number of ways to form 4-digit numbers from given digits = 580.

Note:
In these types of questions use the permutation concept. Permutations are for lists where order matters and combinations are for groups where order doesn’t matter.