A troublesome Counting Problem I have the following question that I fail to comprehend
Mallory is planning to go to Russia in 2018 to see the FIFA World Cup.
Out of the total 64 matches that are being held in the World Cup, she
decides to choose from the first 20 in the first week (she will be in
Russia for only a week), none of which are scheduled at the same
time. There are 4 matches to choose from for each day of the week,
Monday through Friday. Mallory decides to buy tickets for 7 randomly
selected matches out of the 20, with all choices equally likely. What is
the probability that she will have one or more matches to attend every
single day between Monday and Friday?
My Attempt to solution
As we know there are 20 choose 7 total ways of picking 7 tickets from 20 available tickets. We also know that there are 4 ways of picking a ticket for each day from Monday to Friday and there are 15 choose 2 ways of picking remaining 2 tickets from remaining 15 tickets. So all this can be represented as
4 * 4 * 4 * 4 * 4 * 105  //(105 is 15 choose 2)

But when we multiply all of them, it becomes 107520 which exceeds the total number of ways 7 tickets can be chosen from 20 i.e 20 choose 7 = 77520. How is this possible? Where am I going wrong?
 A: Consider the schedule where Mallory attends the matches $1$ and $2$ on both Monday and Tuesday, and match $1$ every other day.
Your count includes this schedule as follows:
match $1$ on Monday through Friday is one of the $4\cdot4\cdot4\cdot4\cdot4$
ways to choose one match each day, and
match $2$ on Monday and Tuesday is one of the $\binom{15}{2}$ ways
to choose the remaining two tickets.
But look: match $2$ on Monday and Tuesday, match $1$ the other three days
is a different choice among the $4\cdot4\cdot4\cdot4\cdot4$ choices for one match each day, and match $1$ on Monday and Tuesday is one of the $\binom{15}{2}$ choices for the remaining tickets. But it is the exact same schedule as the one in the previous paragraph!
In short, you have allowed Mallory to choose the same schedule in several
different ways, and you have counted each of these as a different schedule.

Some hints about a  correct way to count the number of ways Mallory can buy tickets are in the comment by Ross Millikan.
