a) The country's prime minister has reached to $40$ candidates for possible mining. If the number of ministries is $20$, how many different ministerial schemes can the prime minister make, provided that no person will be called upon to take over one ministry?
b) A Megara toll worker is asked to choose his five days
week in which to work. The only limitation is that it must be at least one day
day off during the weekend. How many different options are there?
c) A group of $9$ diabetics visited a café. $2$ people ordered $\text{Greek coffee}$, $3$ people ordered $\text{Espresso}$ and $4$ people ordered $\text{Cappuccino}$ (of course without sugar). The waiter was confused with the order and simply served the drinks to randomly to the members of the group. What is the possibility that the waiter's confusion may be unnoticed, namely serve each member of the group with the drink he ordered?
I want to know if there is a technique to use on these problems or some way to understand what formula to use. Nevertheless my try:
a) I used the combination's formula cause we choose $20$ ministerial spots for $40$ people so ${40\choose 20}=137846528820$
b) If he gets the day off at saturday then he has $6^5$ different ways to arrange his work schedule. If he gets the day of at all weekend then he has only one way and that's it Monday-Friday.
c) I have no idea how to handle this problem.
Thanks for your time.
Best Answer
In question (b), choosing to take Saturday off gives $\binom 65=6$ options (not $6^5$), and there are similarly $6$ option if choosing to take the other weekend day off, and $1$ of those options is common to both schemes (taking the whole weekend). This is a simple example of the inclusion-exclusion principle.
In question (c), you are effectively asking how many anagrams of $GGEEECCCC$ there are (say $q$). Only one of those will be correct, giving a probability of $1/q$ that a random distribution of drinks will be correct. This can be quickly established with a multinomial coefficient, or calulated by multiplying two binomial coefficient successively placing say $G$s and $E$s.