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?

279,936 ways

Step-by-step explanation:

Every day the student has to chose a sandwich from the pile of 6 sandwiches.  So this means the student has to make a choice from the 6 sandwiches for the 7 days. Since the order matters, this is a problem of permutations.

Daily the student has the option to chose from 6 sandwiches. So this means, for 7 days, he has to make a choice out of 6 options. Or in other words we can say, the student has to make selection from 6 objects 7 times.

So, the total number of ways to chose the sandwiches will be 6 x 6 x 6 x 6 x 6 x 6 x 6 =

Alternate Method:

Since the repetition can occur in this case, i.e. a sandwich chosen on one day can also be chosen on other day, the following formula of permutations ca be used:

Number of ways =  

where n is the total number of choices available which is 6 in this case and r is the number of times the selection is to be made which 7 in this case. So,

The number of ways to chose a sandwich will be = ways

check above link.

graphically?   well, you graph them both, the solutions is where they both intersect each other, since there are two solutions, meaning they will meet twice, or namely the linear will run into the quadratic curve twice, and where they meet, those x,y coordinates are the solutions to the system.