Question

4. Every day a student randomly chooses a sandwich for lunch from a pile of wrapped sandwiches. If there are six kinds of sandwiches, how many different ways are there for the student to choose sandwiches for the seven days of a week if the order in which the sandwiches are chosen matters?

Short answer

There are$279,936$ different ways for the student to choose sandwiches for the seven days of a week if the order in which the sandwiches are chosen matters.

Step-by-step solution

Step 1: Definitions

Definition of Permutation (Order is important)

No repetition allowed:$P(n,r)=\frac{n!}{(n-r)!}$

Repetition allowed:$n^T$

Definition of combination (order is important)

No repetition allowed:$C(n,r)=\left(\begin{array}{l}n\\ r\end{array}\right)=\frac{n!}{r!(n-r)!}$

Repetition allowed:$C(n+r-1,r)=\left(\begin{array}{c}n+r-1\\ r\end{array}\right)=\frac{(n+r-1)!}{r!(n-1)!}$

with$n!=n(n-1)\cdot .....\cdot 2\cdot 1$

Step 2: Solution

The order of the elements matters (since we want to select the elements in order), thus we need to use the definition of permutation.

There are six kind of sandwiches and we select for seven days in a week

We are interested in selecting r = 7 elements from a set with n = 6 elements.

Repetition of elements is allowed (else student won’t have any sandwich on the seventh day).

$n^r=6^7=279,936$

Thus there are $279,936$ways in which seven elements can be selected in order from a set with six elements when repetition is allowed.