Wikipedia:Reference desk/Archives/Mathematics/2009 November 22

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November 22

Letter Combinations

I began wondering how many total ways is there to combine six letters without repeating the same thing in a different order?

For example: A,B,C,D,E,F is one possibility. A,A,A,A,A,A is one possibility. A,A,A,A,A,B is another. B,A,A,A,A,A is merely a repeat of the above. Thus, not a possibility.

I could write a program to figure this out, but I thought I learned in high school a way to figure this out using formulas. I think it is by using the second part of http://en.wikipedia.org/wiki/Combination. Nkot (talk) 00:43, 22 November 2009 (UTC)

There's a far fewer number of ways of arranging 6 letters when all you have to work with is 6 letters, than 6 letters out of 26 or more. Your examples don't go beyond F, so maybe you're only looking at 6 in total, but it's not entirely clear. Maybe using 6 objects rather than 6 letters would make it clearer. When you say "different order", would you consider AAAAAB and AAABAA to be the same result, but just in a different order - or is it only mirror images that you call the same order? -- JackofOz (talk) 01:14, 22 November 2009 (UTC)

I see that I was not very clear. I am talking about choosing 6 letters out of 26 letters in such a way that the letters can be repeated. However, each combination should be unique when the letters within it are ordered alphabetically. The fourth example would be the same as the third example if its letters were ordered alphabetically. Your example as well would not be unique. Thus, neither are possible combinations. Nkot (talk) 01:34, 22 November 2009 (UTC)

You can think of the problem as finding how many ways there are of dividing 6 beads among 26 different bins. We can think of having 25 movable dividers that separate the 26 bins, and we can arrange the dividers and beads in whatever order we want. There are 31 objects and 6 of them are beads, which allows ${31 \choose 6}$ different arrangements. See Stars and bars (probability). Rckrone (talk) 02:29, 22 November 2009 (UTC)

I didn't see the part at the end of the question about the formula at Combination#Number of combinations with repetition. You're right about that being the correct formula. They also explain the stars and bars thing there in different words. Rckrone (talk) 04:16, 22 November 2009 (UTC)

Max symbol

I was looking in Table of mathematical symbols and I don't see a symbol to indicate the maximum value in a set (or the maximum value between two values). I want to write: absolute value of the length of A minus the length of B <= D <= the maximum between the length of A and the length of B. What I have is: ||A|-|B|| <= D <= max(|A|,|B|). I don't like the mixed use of | for both "length of a vector" and "absolute value". I also want to avoid computer programming appearance with max(). -- kainaw™ 05:03, 22 November 2009 (UTC)

Do you mean using $A\wedge B$ for min and $A\vee B$ for max? (Igny (talk) 05:18, 22 November 2009 (UTC))

Those are far from standard notations. --Tango (talk) 07:06, 22 November 2009 (UTC)

Actually $A\wedge B$ and $A\vee B$ for inf and sup of $\{A,B\}$ didn't meet a great success outside the theory of ordered structures and lattices (where they are a standard notation, however). Most likely, the reason is that the symbols $\wedge$ and $\vee$ are already largely and universally used for operations in the exterior algebras. --pma (talk) 09:22, 22 November 2009 (UTC)

They may be obsolete in many fields of math, but I have seen their usage in a number of recent publications on measure theory and probability theory. (Igny (talk) 00:14, 23 November 2009 (UTC))

See for example here, page 4, formula 1.2 (Igny (talk) 01:12, 23 November 2009 (UTC))

The mathematical term for the largest element of a set is supremum, abbreviated sup. But I think that is not what you are asking. We'd usually call the "length of a vector" its norm rather than its absolute value. The norm is denoted $\|x\|$. So you'd write

$|\|A\|-\|B\||\leq D\leq \max(\|A\|,\|B\|)$.

67.117.145.149 (talk) 06:48, 22 November 2009 (UTC)

The largest element in a set is the maximum. The least upper bound of a set is the supremum. They are different things. If a set has a maximum that will also be its supremum, but a set can have a supremum without having a maximum since the least upper bound is not necessarily a member of the set. For example, $\sup[0,1)=1$, but that set has no maximum. --Tango (talk) 06:51, 22 November 2009 (UTC)

Oh yes, good point. 67.117.145.149 (talk) 06:57, 22 November 2009 (UTC)

The standard mathematical notation for a maximum is "max", whether that looks like computer code or not. --Tango (talk) 06:51, 22 November 2009 (UTC)

The J (programming language) uses >. for max and <. for min.

   3 >. 4
4

Bo Jacoby (talk) 08:02, 22 November 2009 (UTC).

Any discontinuous function with standard notation will reduce your problem to one already solved, as long as you define its value at the discontinuity as half-way between the values its jumping between. For example,

$\max\{a,b\}={\frac {a}{2}}({\mbox{sgn}}(a-b)+1)+{\frac {b}{2}}({\mbox{sgn}}(b-a)+1)$.

Expz (talk) 00:51, 23 November 2009 (UTC)

If we wanted to write max in a silly longwinded confusing way, we could write $\max\{a,b\}={\frac {a+b+|a-b|}{2}}$. Algebraist 01:56, 23 November 2009 (UTC)