Question

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

Answer 1

Given:

The numbers are $1,2,3$

To find:

The number of combination's without repeating the given number.

Explanation:

We know that, the combination is defined as "for $n$ things choosing $r$ has combinations.

${}^{n}C_r=\frac{n!}{r!\left(n-r\right)!}$

$3$ choose $1$ is we get as

$\begin{array}{rcl}{}^{3}C_1& =& \frac{3!}{1!\left(3-1\right)!}\\ & =& \frac{3\times \cancel{2!}}{1\times \cancel{2!}}\\ & =& 3\end{array}$

$3$ choose $2$ is we get as

$\begin{array}{rcl}{}^{3}C_2& =& \frac{3!}{2!\left(3-2\right)!}\\ & =& \frac{3\times \cancel{2!}}{\cancel{2!}\times 1}\\ & =& 3\end{array}$

$3$ choose $3$ is we get as

$\begin{array}{rcl}{}^{3}C_3& =& \frac{3!}{3!\left(3-3\right)!}\\ & =& \frac{\cancel{3!}}{\cancel{3!}\times 1}\\ & =& 1\end{array}$

Now by adding all the combinations we get

$\begin{array}{rcl}{}^{3}C_1+{}^{3}C_2+{}^{3}C_3& =& 3+3+1\\ & =& 7\end{array}$

Therefore, there are $7$ ways of combination for the number $1,2,3$.