FCS Practice Session 1.
September 8, 2020
Basic counting rules
[Product Rule] If something can happen in n1 ways, and no matter how the first thing happens, a second thing can happen in n2 ways, and so on, no matter how the first k−1 things happen, a k-th thing can happen innk ways, then all thek things together can happen inn1×n2×. . .×nk ways.
[Sum rule] If one event can occur in n, ways, a second event can occur in n2 (different) ways, and so on, ak-th event can occur innk (still different) ways then (exactly) one if the events can occur in n1+n2+. . .+nk ways.
A permutation of a set of n elements is an arrangement of the elements of the set in order. The number of permutations of an n-set is given byn×(n−1)×. . .×1 =n! (product rule).
Given an n-set, suppose that we want to pick out relements and arrange them in order. Such an arrangement is called an r-permutationof then-set. The numberP(n, r) orr-permutations of ann-set is given by n×(n−1)× . . .×(n−r+ 1) (product rule).
Anr-combination of ann-set is a selection ofrelements from the set. Order does not count. (i.e., anr-combination is an r-element subset.) nr
will denote the number of r-combinations of ann-set. Notice thatP(n, r) = nr
×r!
(product rule) and so nr
=r!(n−r)!n! . We have: nr
= n−1r−1
+ n−1r .
If we are choosing an r-permutation out of an n-set with replacement then we say that we are sampling with replacement. The product rule gives us that the number ofr-permutations of an n-set with replacement isnr. Similarly, we may speak of r-combinationsof ann-set with replacement or
repetition. For example, the 3-combinations of a 2-set {0,1} with replacement are {0,0,0}, {0,0,1}, {0,1,1}, {1,1,1}.
The number of r-combinations of ann-set with repetition is n+r−1r .
Order Repetition The sample Number of ways to counts? allowed? is called: choose the sample:
No No r-combination nr
Yes No r-permutation P(n, r) = (n−r)!n!
No Yes r-combination n+r−1r
with replacement
Yes Yes r-permutation nr
with replacement [Binomial Theorem] Forn≥0, (a+b)n=Pn
k=0 n k
akbn−k .
1. Before the licence plate reform the form of the Hungarian licence plates was LL-DD-DD (L=letter, D=digit).
How many different licence plates were possible? How muany more can we have now, with the new system of LLL-DDD? The Dutch licence plates are of the form XX−Y Y −ZZ where {X, Y, Z} ={L, D}. How many possible plates are there in The Netherlands?
2. How many subsets does an n-element have? How many 0/1-sequences of lengthn are there? How many 0/1-sequences of lengthnare there that contain exactlyk 1’s?
3. How many matches are played if n soccer teams play a round-robin tournament to decide the best team?
How about if straight elimination is used?
4. How many ways can we fill up a 5 out 90lottery ticket? If we fill up lottery tickets in all possible ways, how many 5-, 4-, 3- or 2 matches tickets do we have after the drawing?
5. How many length 10 rolling sequence of the ordinary die are there that the sum of the numbers rolled is divisible by 3?
6. How many rooks can be placed on the chessboard so that none of them hits the other? How many ways can the maximum number of rooks be placed on the chessboard?
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7. Prove that for all positive integern, equalityPn i=1i ni
=n·2n−1 holds.
8. How many 5 digit numbers are there that contain digit 3?
9. It is a scientifically proven fact that the flags of countries on continent Atlantis consist of three horizontal stripes, which can be of colors red,white, green, blue, yellow or black, such that stripes next to each other have different colors. Different countries have different flags, of course. How many countries can be on Atlantis?
How many such countries can be that have red stripe in their flags?
10. Eight people want to play tennis on three courts so that on one court they play doubles and on the other two courts they play singles.How many ways can this be done, if the courts are distinguished, but the sides of the courts are not.
11. How many ways can 30 policemen be partitioned into six patrols of size five?
12. There arenrows of chairs in a movie theater with k1, k2, . . . , kn chais, respectively. How many ways canm people be seated in the theater? How many ways can ` married couples be seated in a row of k chairs, if wifes must sit next to their husbands?
13. How many ways are there to seat n2 people inn rows of chairs of nchairs each, if in each row the people must sit in increasing order of their age, from left to right (Let us assume that the ages of people are pirwise distinct)?
14. All 556 electrical engineering students wrote two midterms, one of FCS and one of Analysis. Nobody got more than 36 points on the midterm of FCS. Prove that there are four students such that besides receiving the same number of points on their FCS midterms, they got the same grade (mark) on their Analysis midterms.
15. Prove that among 9 lattice points inR3(that is points of integer coordinates) there exists two, such that the midpoint of the segment connecting them is also a lattice point.
16. How many ways can we get from the corner of 14th Street and 10th Avenue in New York City to the corner of 23rd Street and 5th Avenue, if we walk on streets and always towards our goal?
17. How many ways can we read out the word METAMATEM- ATIKATEMATIKAin the following table if we start from the upper left corner and always go right or down?
M E T A M A T E M A T I K A
E T A M A T E M A T I K A T
T A M A T E M A T I K A T E
A M A T E M A T I K A T E M
M A T E M A T I K A T E M A
A T E M A T I K T E M A T
T E M A T I K A T E M A T I
E M A T I K A T E M A T I K
M A T I K A T E M A T I K A
18. How many ways can we get a small rectangular piece from a sizen×kchocolate bar if we can only break it along perforation?
19. 10 burglars keep their stolen treasure in a safe with a huge number of locks. They want to put on locks and distribute keys so that any 4 burgler could open the safe together, but no three of them have all the necessary keys. How many locks do they need? How many locks do they need if they want that the Chief and any other burglar could open the safe together (but the Chief alone could not do it), besides the rule above (*)?
20. How many ways can 9 members of the monetary committee of the parliament can be chosen from 18 delegates, if at most 3 of the 7 delegates of the opposition parties can be selected?
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