Permutation - Practice Questions & MCQ

Permutation basically means the arrangement of things. And when we talk about arrangement then the order becomes important if the things to be arranged are different from each other (when things to be arranged are the same then order doesn’t have any role to play). So in permutations order of objects becomes important.

Arranging n objects in r places (Same as arranging n objects taken r at a time) is equivalent to filling r places from n things.

So the number of ways of arranging n objects taken r at a time = n(n - 1) (n - 2) ... (n - r + 1)

$$
\frac{n(n-1)(n-2) \ldots(n-r+1)(n-r)!}{(n-r)!}=\frac{n!}{(n-r)!}={ }^n P_r
$$

Where $r \leq n$ and $r \in W$

So, the number of ways arranging n different objects taken all at a time = ${ }^n P_n=n!$.

Example: In how many ways can 5 people be seated at 3 places?

Solution: Basically this question is about arranging 5 people at 3 different places

    Let’s think that we are given 3 places, so for the first place we have 5 people to choose from, hence this can be done in 5 ways as all 5 are available.

    Now for 2nd place we have 4 people to choose from, hence this can be done in 4 ways.

    Similarly, for 3rd place, we have 3 choices.

    Since we have to choose for all 3 places, the multiplication rule is applicable, and the total number of ways  5×4×3=120, 

    This can also be done directly from the notation or formula

    ${ }^{\mathrm{n}} \mathrm{P}_{\mathrm{r}}$ where $\mathrm{n}=5, \mathrm{r}=3$, so ${ }^5 \mathrm{P}_3=\frac{5!}{2!}=5 \times 4 \times 3=120$

Example: Find the number of ways the letters of the word “BIRTHDAY” can be arranged taken all at a time.

Solution: From the above concept directly using the formula we have 

${ }^8 \mathrm{P}_8=8!=40,320$