Download MAT 243 Test 3 Practice

MAT 243 Test 3 Practice Solutions
1. (6+3)/ C(11,3)
2. Use PHP. N=? k = # of couples =n ceiling(N/k)=2. Thus N = (2-1)n+1 = n+1.
3. C(7,4)/2^8
4. Use GPHP. N=? k = # of months =12 ceiling(N/k)=4. Thus N = (4-1)12+1 = 37
5. 9^n = (8+1)^n [use the binomial theorem]
6. A bag contains 100 apples, 100 bananas, 100 oranges and 100 pears. If someone randomly picks a fruit
out of the bag every second, how long will it be before she is assured of having at least a dozen pieces
of fruit of the same kind?
(a) How long will it be before she is assured of having at least a dozen bananas? Use GPHP . N=?
k=4 (types of fruits) ceiling(N/k) = 12. N=(12-1)*4+1 =45 seconds
(b) In the worst case scenario you picked all the other fruits before the 12th banana, so 312 seconds.
Note: this is a common sense solution, absolutely no connection to GPHP.
7. Let A be the set of all ternary strings of length 9 that start with a 1, B be the set of all ternary strings of
length 9 that end with a 2, and let C be the set of all ternary strings of length 9 that has a 0 in the middle
position. How many ternary strings of length 9 are there that either start with a 1, end with a 2 or has a 0
in the middle position?
Use Inclusion-Exclusion Principle
3^5. So |ABC|= 4617
8.
|A| = |B| = |C|= 3^7 . |AB|=|AC|=|BC|=3^6 and |ABC| =
Prove that in any group of n people, there are at least 2 who have shaken hands with the same number
of people. (Shaking hands with yourself does not count)
Use PHP. N= # of people = n. k = # of hands shakes. The possibilities are {0,1,…,n-1} BUT notice
that if someone was shaking hands with n-1 people means he was shaking EVERYBODY’s hand, so
then 0 is an impossible answer for all other people and similarly, if someone was shaking hands with 0
people means he was shaking NOBODY’s hand, so then n-1 is an impossible answer for all other
people. Therefore k= n-1. Thus, ceiling(N/k) = 2 which completes the proof.
9. In a box there are 6 white, 5 blue and 4 red marbles. If we pick a total of 5 marbles without replacement,
how many different ways can we pick
(a)
exactly 3 white marbles? C(6,3)C(9,2)
(b)
at least 2 blue marbles? C(5,2)C(10,3)+ C(5,3)C(10,2)+ C(5,4)C(10,1)+ C(5,5)C(10,0) or
C(15,5)- C(5,1)C(10,4)+ C(5,0)C(10,5)
(c)
exactly 2 white marbles and 1 blue marbles? C(6,2)C(5,1)C(4,2)
10. How many different 10 letter words contains the string NICE if letters can be repeated?
Think of “NICE” as a super letter. Then we have 6 more letters to choose, each can be 26 ways and
NICE can be at 7 different position. Answer = 726^6
11. How many 10 digit numbers contains the string 123 if numbers cannot be repeated?
Answer = 8!
12. There are 100 people at a party. Each person has an even number (possibly 0) of acquaintances. Prove
that there are three people at the party with the same number of acquaintances. USE GPHP. N = # of
people, k = # of acquaintances. Possible k values are {0,2,4,....,98}. Just like in #8 if two people has 0
acquaintances than nobody has 98 acquaintances. Or if two people have 98 acquaintances than at most
one person can have 0 acquaintances. In the first case N=100 and k = 49, in the second case N = 99 and
k = 49. Then ceiling(N/49)=3 in both cases.
13*. Prove that any given 52 integers, there exist two of them whose sum or else whose difference is
divisible by 100.