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8h
comment What is the probability that a customer will not use a credit card? Pays in cash or with a credit card?
The reasoning is well described, and correct.
9h
comment Inequtions problem - how to calculate total of sales for a determined ROI?
The gross income is $220x$. So the details are not at all right.
9h
revised Opposite of Fermat's Last Theorem?
added 11 characters in body
9h
answered Opposite of Fermat's Last Theorem?
9h
comment Trying to construct a specific function
No reason, I just like geometric progressions to start at $1$.
9h
answered Trying to construct a specific function
11h
comment Multinomial Coefficients
I do not think there is a natural approach through a multinomial coefficient.
12h
comment How many $2\times3$ real matrices are needed to guarantee that at least one of them is a linear combinations of the others?
Given that the answer is $7$, can you see why?
13h
comment Since $\lim\limits_{x\to0}\frac{\sin kx}{kx}=1$ for constants $k$, is it also true for general arguments?
Yes, if $f(x)$ has limit $0$ then $\frac{\sin(f(x))}{f(x)}$ has limit $1$.
13h
comment Combinatorics: How many 6 digit numbers have AT LEAST one '9' among them?
Exactly $5$ nines means we fill the places with nines, only one way to do it. Exactly $4$ means we choose the places where nines will go, $\binom{5}{4}=5$ ways to do it, then put anything other than nine in the remaining spot, total $(5)(9)$. Exactly $3$ nines means we choose $3$ places, $\binom{5}{3}=10$ ways, fill the remaining two spots with non nines, total $(10)(p^2)$. And so on.
14h
comment How many ways to select distinct pairs from k disjoint sets
You are welcome.
15h
comment How many ways to select distinct pairs from k disjoint sets
If you are interested in more elaborate identities of this type, look for elementary symmetric functions.
15h
comment How many ways to select distinct pairs from k disjoint sets
The identity $2\sum_{i\lt j}x_ix_j=(x_1+\cdots +x^n)^2-(x_1^2+\cdots+x_n^2)$ comes up pretty often, particularly in Statistics. It is a generalization to $n$ variables of the familiar $2xy=(x+y)^2-(x^2+y^2)$. What I gave is not really a closed form, since it still involves $\sum$. Note that for $n=3$ your way is more efficient. But for example when $n=100$, the formula I suggested is far more efficient.
15h
comment Permutations and Combinations - conceptual
Yes, the string of comments had reached unreasonable length, so we both deleted..
15h
revised How many ways to select distinct pairs from k disjoint sets
added 69 characters in body
15h
answered How many ways to select distinct pairs from k disjoint sets
15h
comment Permutation of students in a class
Do you mean two specific MS students, or some two MS students? In neither case is $\frac{16!}{5!10!}$ right, and two specific would not do the job. One form of the correct answer to the original problem is $\binom{11}{7}10!7!$. There are many "no two type X people sit together" questions on MSE.
17h
comment Combinatorics: How many 6 digit numbers have AT LEAST one '9' among them?
Your suggested approach is good. More work, but not terribly awful. First count the numbers whose first digit is $9$. Easy. Then we want the number of numbers with something other than $9$ in first position. There are $8$ choices for first digit. For each choice, count the number of $5$-digit strings with at least one $9$. Exactly $5$ $9$'s, easy. Exactly $4$, $\binom{5}{4}9^4$. And so on. By the way, your simple expression is right, but I think $472,392$ is not.
18h
comment Expected value of a biased coin toss
I think you are asking about the expectation of a random variable $X$ that has geometric distribution with parameter $1-p$ (unusual choice of name). We have $E(X)=\frac{1}{1-p}$. The expectation of a geometric random variable, and sums of the type you are interested in, have often been discussed on MSE.
19h
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