How many variances of a working day in a week

Question

Go easy on me please, have been scratching my head and googling all day to try and figure out the solution to this. If I have employees that can work $1$ or $2$ or $3$ or $4$ or $5$ days a week, how many variances of working day patterns are there? So for $1$ day a week worker, there are $7$ variances (they can work mon/tues/wed/thurs/etc...) for $2$ day a week worker, there are $21$ variances (I think) I've worked out long hand. How many for each type of worker? Thanks!

Answer 1

HINT:

You could think as follows

For the $1$ day case you have a bowl called the work day and you would like to count the different possibilities to fill it, and you are right it is $7$ different possibilities. You could also think as how many ways to pick one object out of $7$ objects which is usually denoted and evaluated as follows $$ C_n^{k}=\binom{n}{k}=\frac{n!}{k!\cdot(n-k)!} $$ So in this particular case $$ C_7^{1}=\binom{7}{1}=\frac{7!}{1!\cdot6!}=7. $$

For the two day variation you may think of it as $2$ bowls which are to be filled with one day each, for the first bowl you could choose out of $7$ different days, and for the second bowl you have $6$ different days left to choose from. For each of the $7$ choices for the first bowl you will have $6$ choices for the second one which gives $7\cdot 6=42$ and we have to divide by $2!$ since we can arrange the bowls in $2$ different ways (you could think of this as we gave an order to the bowls first and second which was not needed).

Or you could go and use the formula giving

$$ C_7^{2}=\binom{7}{2}=\frac{7!}{2!\cdot(5)!}=21. $$

I hope this helped