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What is the probability of the sum of four dice being 22?

Question Four fair six-sided dice are rolled. The probability that the sum of the results being $22$ is $$\frac{X}{1296}.$$ What is the value of $X$? My Approach I simplified it to the equation ...
1k views

Book on combinatorial identities

Do you know any good book that deals extensively with identities obtained using combinatorial and/or probabilistic arguments (e.g., by solving the same combinatorial or probability problem in two ...
2k views

Number of occurrences of k consecutive 1's in a binary string of length n (containing only 1's and 0's)

Say a sequence $\{X_1, X_2,\ldots ,X_n\}$ is given, where $X_p$ is either one or zero ($0 < p < n$). How can I determine the number of strings, which do contain at least one occurrence of ...
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Probability event with $70$% success rate occurs three consecutive times for sample size $n$

It has been a long time since I've done probability so I am not sure which to do (if either are correct). Thank you for taking the time to look at my work. Probability an event occurs is $70$%. I'm ...
651 views

Number of solutions of $x_1 + x_2 + x_3 + x_4 = 14$ such that $x_i \le 6$

Let $x_1, x_2, x_3, x_4$ be nonnegative integers. (a) Find the number of solutions to the following equation: $$x_1 + x_2 + x_3 + x_4 = 14$$ I got $17 \choose 3$ for this. Is ...
346 views

Number of q-ary strings of length m which do not contain k consecutive zeros

A finite q-ary-alphabet is given $$A_q = {0,1,2,...,q-1}.$$ Now I am considering the set of all finite strings over the alphabet $A_q$. I am interested on the number $$N(m,k)_{A_q}$$ of strings of ...
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Efficient algorithm to find all unique combinations of set given duplicates

The number of combinations for a 4 choose 2 is 6. But what if I have a set in which 2 of the elements are the same? Of course I could do a unique check for each item but this is too computationally ...
743 views

Number of solutions of $x_1+x_2+\dots+x_k=n$ with $x_i\le r$

Let $n,k,r$ be positive integers. The number of all nonnegative solutions of the Diophantine Equation $x_1+x_2+\dots+x_k=n$ is $\binom{n+k-1}{n}$. Is there a general formula for the number of ...
472 views

Number of combinations with sets of identical items

I was trying to write a python program which takes a string and prints out all possible combinations of characters from it and in turn figure out the number of such possible combinations of a string ...
217 views

Number of bit strings of length $n$ with no $k_1+1$ consecutive 0s and no $k_2+1$ consecutive 1s.

Just as the question asks. I am trying to calculate the number of bit strings of length $n$ with a maximum of $k_1$ consecutive $0s$ and $k_2$ consecutive 1s. Of course we assume $k_1+k_2\leq n$. I am ...
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How do you calculate the sum of combinations of 1000 dice rolls?

For two dice rolls we can calculate the number of combinations for each summed total: Rolling a 2: one chance (1&1) Rolling a 3: two chances (2&1)(1&2) Rolling a 4: three chances (3&1)...
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Probability of rolling a sum of at least 18 with 10 6-sided dice

I'm trying to work out how to do this, and I'm very stuck. My grasp of probability is shaky to begin with, and generally with probability questions, I list out cases. Due to the number of ...
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How to calculate dice roll probability with more dices? [duplicate]

I'm not math graduate, so be patient. For two dices it is easy, because of small number of possibilities, also it can be easy for three, but how is it works for ten (some formula)? How to calculate ...