Question

How do you tell if a word problem is a permutation or combination $?$

Answer 1

Defining permutation or combination:

Permutation is all about the arrangements of objects/elements, whereas combination is about the selection of objects/elements. In permutation, order of elements/objects matters, whereas in the combination order of elements/objects does not matter.

Example 1: How many different teams of $6$ students can be chosen from $11$ students.

Therefore, Total number of teams of $6$ students $={}^{11}C_6$ $\left[\mathbf{\because }{}^{\mathbf{n}}\mathit{C}_{\mathbf{r}}\mathbf{}\mathbf{=}\mathbf{}\frac{\mathbf{n}\mathbf{!}}{\left(n-r\right)\mathbf{!}\mathbf{r}\mathbf{!}}\right]$

$$\begin{gathered} =\frac{11!}{\left(11-6\right)!6!} \\ =\frac{11\times 10\times 9\times 8\times 7\times 6!}{5!6!} \\ =\frac{11\times 10\times 9\times 8\times 7}{5\times 4\times 3\times 2\times 1} \\ =262 \end{gathered}$$

Example 2: In how many different ways $6$ students can be seated in $10$ places.

Therefore, Total ways of seating arrangement of $6$ students in $10$ places $={}^{10}P_6$ $\left[\mathbf{\because }{}^{\mathbf{n}}\mathit{P}_{\mathbf{r}}\mathbf{}\mathbf{=}\mathbf{}\frac{\mathbf{n}\mathbf{!}}{\mathbf{(}\mathbf{n}\mathbf{-}\mathbf{r}\mathbf{)}\mathbf{!}}\right]$

$$\begin{gathered} =\frac{10!}{\left(10-6\right)!} \\ =\frac{10\times 9\times 8\times 7\times 6\times 5\times 4!}{4!} \\ =151200 \end{gathered}$$

Hence, by above conditions we can tell if a word problem is a permutation or combination .