Question

How many combinations can you make with the number $1,2,3$?

Answer 1

Though this type of sum can be solved easily it would be great to to use the formula of Combinations to find the Answers. This is because if a student has the hang of using the formula for combinations and permutations it would become extremely easy for him to solve questions on probability, series numerical. In this particular numerical we would be using the formula $^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}$ to find the number of combinations.

Complete step-by-step answer:
In order to solve the numerical, we can say that the 3 digits need to be rearranged in any manner to find the number of combinations possible.
Thus we can say that for thousands of places the number of combinations possible are $^3{C_1}.......(1)$. This is because there are $3$ numbers and only $1$ blank space is available.
Now similarly finding the number of combinations for $100s$ place. Now since $1$ digit has been used up in the thousands place, we are left with $2$ digits and $1$ blank space. Therefore the numbers of combinations possible are $^2{C_1}........(2)$.
Now we are left with only $1$ digit and we have only $1$ blank space. The number of combinations possible are $^1{C_1}........(3)$.
In order to find the total number of combinations, we will have to multiply the equation $1,2,3$.
The total number of combinations are $^3{C_1}{ \times ^2}{C_1}{ \times ^1}{C_1} = 3 \times 2 \times 1 = 6$.
The list of combinations are as follows
$\begin{gathered} 123 \\\ 132 \\\ 213 \\\ 231 \\\ 312 \\\ 321 \\\ \end{gathered}$

Note: Though this sum looked easy, it could become tedious if there were more numbers of digits and we would find total combinations by just thinking and writing it down without the usage of the formula. For example, if there was a sum in which $6$ digits were given and we had to find the number of combinations, it would be a $1$ minute sum if we used the formula, on the contrary it would take a lot of time if we found the answer without any formula. Thus the student should start practicing the use of the formula.