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### I roll a die $N$ times, what is the probability that the sum is exactly $k$? [duplicate]

The die is six-sided. So, this is easy with a two-sided coin showing exactly $k$ heads, as that is just a Bernoulli trial being run $n$ times (Binomial RV). You can also approximate this using a ...
17k views

### 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 of ...
2k 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 ...
3k views

### 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 ...
4k 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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### 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 ...
461 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 ...
741 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 ...
1k 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 ...
737 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 ...
314 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 ...
4k views

### How to calculate dice roll probability with more dice? [duplicate]

I'm no math graduate, so be patient. For two dice it is easy, because of the small number of possibilities, and itâ€™s still easy for three, but how can I work out the case for ten (with some formula)? ...
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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)...