A cashier’s till has a large number of pennies, nickels, dimes, and quarters. In how many ways can 10 coins be chosen from the till?
27844
I have a problem that I am unable to solve. I spent too much time on this one and I think I miss something.  An ice-cream-shop has 11 flavors of ice-cream. In how many ways can one person choose 6 cornets with flavors not necessarily different?
37252
A student prepares for an exam by studying a list of 12 problems. She can solve 9 of them. For the exam, the instructor selects 7 problems at random from the 12 on the list given to the students. What is the probability that the student can solve 5 of the 7 problems on the exam?  I am really confused by this problem. My first initial thought was to do:  (5 chose 7)(7 chose 0)/ (12 chose 7)  I am not 100% sure about this though.
38225
I'm not sure how well known this "riddle" is but here it goes.  3 people go to a restaurant, each buy food worth 10.00.  When they're done, they give 30.00 to the waitress.  She gives the money to the manager, manager says they paid too much, gives 5.00 back to the waitress.  She comes back to the table, gives a dollar back to each person and then puts the remaining 2.00 in her pocket for tip.  So now, each person has paid 9.00.  9.00 x 3 = 27.00.  Plus the 2.00 in the waitresses pocket is 29.00.  What happened to the 30th dollar?  So what's the issue with this way of calculating (since if you backward it works fine), that doesn't give us the correct result?
58639
Question: Assume you choose any $48$ positive integers so that the number of different prime factors of all those numbers is $10$. Prove or disprove there exists a set of four numbers out of the $48$ positive integers of which product is a perfect square.   I think that it uses pigeon hole, but I couldn't make a progress.
66702
I was solving some combinatorics question and I came upon one I've been trying to solve for 2 days and I'm curios how to solve it.  It is known that in every set of 15 cars, at least 3 were manufactured from the same country (there can also be another 3 (or more) cars in the set which came from another country). prove that in every set of 100 cars, 15 were made manufactured in the same country.  Thanks to anyone who helps!
75749
0 choices for the 1st person.  17 choices for the 2nd person (must exclude 1st and his/her two neighbours)   For 2 of these choices of 2nd person, there is one shared neighbour, so 15 remaining choices. (e.g. if they are numbered 1 to 20 in a circle, 1st person is #1, 2nd is #3, then people 20,1,2,3,4 are excluded).  For the other 15 choices of 2nd person, there are no shared neighbors, so 14 remaining choices.   So if order matters, total is $20 \cdot (2 \cdot 15 + 15 \cdot 14) $ but since order does not matter, divide by $3! = 6$ to account for the permutations in order of the 3 people.  So total = $20 \cdot \frac{2 \cdot 15 + 15 \cdot 14}{6}$  just redid it; does this make any sense?
26852
A friend of mine asked me the following question:      Suppose you have a basket in which there is a coin. The coin is marked   with the number one. At noon less one minute, someone takes the coin   number one and put into the basket two coins: the number two and the   number three. At noon less $\frac{1}{2}$ minute, he takes the coin   number two and put into the basket the coins number $4$ and the number   $5$ and so on. At noon less $\frac{1}{k}$ he takes the coin number $k$   and put two other coins.       The question is: How many coins I can find in the basket at noon?   Intuitively the answer is $\infty$ because I added one coin for infinite times. But the correct answer seems to be $0$ because every coin numbered $k$ has been removed at noon less $\frac{1}{k}$ of minute. What is the correct answer? Thanks.
110455
Consider a coin that can move in at most two ways when it is placed on any square of the usual (8 x 8) chessboard (64 squares). From its current square it can move either to the adjacent square above (if it is not in the top most row) or to the adjacent square on the right (if it is not in the rightmost column).   What is the number of distinct paths in which the coin can move from the lower left corner square to to the top right corner square ?  I understand the problem and it should be solved using recursion technique but every time I approach this kind of problem I take lot of time. Sometimes I get the same problem in different flavor which says it can only move down or left(any combination of left/right/up/down).  Can someone give me some ideas how to start with these problem?  I found answer to be 688 which may be wrong.   Help appreciated :)
89450
I'm stuck with the basic counting problem.  Problem is the following:     Corrupt professor Z has a class of 50 students. He needs to give exactly 10 A's. However five students already have a special deal (they are professor Z's nephews and neices) and will get A's for sure. How many ways can the 10 A's be distributed?   My thought : 45 Choose 5, because we can ignore 5 people(nephews and nieces) who will get A's grade for sure.   I'm not sure my thought is correct or not. I'm appreciated any comment or explanation for this.
94167
Let's suppose there is a chessboard (with 64 squares), and there is a penny on each square. Each of the pennies has a side (head or tails) facing up, chosen at random.  I look at the board and choose my favorite penny. I then tell my two mathemetician friends that I will reveal which penny is my favorite to one of them, who then must flip over exactly one of the pennies on the chessboard, while the other is in another room not looking. Once the coin is flipped over, the uninformed mathematician is summoned into the room and must deduce which penny is my favorite only by examining the coins on the chessboard.  What strategy can the two mathematicians devise to ensure that my favorite penny can always be deduced?
30844
If $n$ distinct things are arranged in a circle, then what are the number of ways selecting three of these things so that no two of them are next to each other?
78249
A village has about $1$ million residents. Suppose each resident has a jar with $100$ coins in it.Two jars are considered to be “equivalent” if they have the same number of 10c, 20c, 50c, $\$$1 and $\$$2 coins in them. How many different (i.e. nonequivalent) jars of coins exist? Is it theoretically possible that every person in this village has a different jar of coins?  I understood that if we have $k_1$ 10 coins in the jar where $k_1$ is between $0$ and $100$ inclusive, then there are $100-k_1$ coins of other denominations in the jar. Likewise, we also have $k_2$ 20c coins in the jar where $k_2$ is between $0$ and $100$ inclusive, so the combined number of 50c, $\$$1, and $\$$2 coins in the jar is $100-k_1-k_2$. And we always have some $k_1, k_2, k_3, k_4, k_5$, let's say, so that the sum of those $k$'s is $100$.  I got stuck since I don't know how to 'compare' the number of coins in jars for different residents.
40571
I pick a random subset $S$ of $\{1,\ldots,N\}$, and you have to guess what it is. After each guess $G$, I tell you the number of elements in $G \cap S$. How many guesses do you need?
43266
I thought of this question lately, but I'm not satisfied with the answer I got:  If I flip a coin 100 times, what is the probability that I will get a streak of at least ten of the same side?  The way I thought of solving it was just: $\left(90 \times 0.5 ^ 9\right) = 0.0879$, but this has to be wrong because, for the probability of getting a streak of at least 4 in the same scenario, it yields: $\left(96 \times 0.5 ^ 3\right) = 12$, which obviously doesn't make sense.
60413
Problem: Players A,B and C toss a fair coin in order, the first to throw a head wins what are their respective chances of winning?  Attempt: Let X = event that A throws a head on the first toss, and Y = event that B throws a head on first toss, similarly let Z = event that C throws a head on first toss.  Then there are eight different triple for tossing a coin.  S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}   where H is a head, and T is a tail.  Can anyone please help me? I don't know how to continue. Any feedback/help would be really appreciated. Thank you.
85075
A friend and I are going back and forth with brain-teasers and I have no idea how to solve this one.  My assumption is that it's possible with some bitwise operators, but not sure.
7970
I a programmer not a mathematician so please excuse my ignorance and please dumb it down for me.  My research indicates that this is a variation of the Coupon Collector problem but I really don't know how to get the answer.  Assuming $n$ equally likely desired selections (Probability for each=$1/m$) how many selections with replacement must be made to have an $X$ probability of getting at least one of each of the $n$ desired objects?