Creating Even Problems with Restrictions: x!

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Discussion Overview

The discussion revolves around creating mathematical problems involving factorials with specific restrictions on the integers included in the calculations. The focus is on two scenarios: one where odd numbers are omitted given that x is even, and another where even numbers are omitted if x is divisible by 3. Participants explore the feasibility of these tasks and the implications of the restrictions.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions the intent behind "making a problem" and suggests that clarity on the intended result is necessary for assistance.
  • Another participant proposes a formula for the product of odd numbers derived from a factorial, specifically using 9! as a starting point.
  • A different participant explains how to express the product of even numbers based on the value of x, providing a formula for both even and odd cases.
  • One participant outlines the implications of omitting even numbers when x is divisible by 3, detailing the resulting odd integers that remain and suggesting a product formula for those integers.
  • Another participant expresses gratitude for the discussion and indicates a potential typo in their original problem statement, suggesting they may need to clarify their question later.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the clarity of the original problem or the best approach to formulate the problems. Multiple interpretations and methods are presented, indicating ongoing uncertainty and exploration.

Contextual Notes

Participants express varying levels of understanding regarding the problem's requirements, and there are indications of potential typos or miscommunications that could affect the discussion's direction.

Who May Find This Useful

Readers interested in mathematical problem creation, factorials, and combinatorial reasoning may find the discussion relevant.

Elihu5991
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I have been given a problem to make a problem for x!. The only restriction is, "But omit odd numbers given x is even"

Another is for x! is, "Omit even numbers, x is divisible by 3".

Is it even possible? I've thought of several manners, but I don't think I'm correct.

Thanks in advance, everyone.
 
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What do you mean "make a problem"? Are you supposed to design a math question?
If I don't know your intended result I cannot help you. How would you know if you are "correct"?

note, if n is a positive integer, then xn=2n-1 is an odd number.
Does that help?
 
Elihu5991 said:
I have been given a problem to make a problem for x!.

What do you mean "make a problem"? Is that a typo?

I think what you are asking for is a formula for, say,
[tex]1 \cdot 3 \cdot 5 \cdot 7 \cdot 9[/tex]
Okay, start with 9!
How do you construct the product above from 9 factorial?
 
those are sometimes denoted (x)!. If x is even it is x(x- 2)(x- 4)...(4)(2). If x is odd, it is x(x- 2)(x- 4)...(3)(1).

Note that is x is even, say x= 2n, then we have (2n)(2n-2)(2n- 4)... (4)(2)= (2)(n)(2)(n-1)(2)(n- 2)...(2)(2)(2)(1)= n(n-1)(n-2)...(2)(1) times a power of 2.
 
Thankyou so much everyone! This problem was a while back. I have had problems accessing this forum, that's why I haven't replied in a long time.

If I can remember what was the problem (I don't normally forget things), as it may have been a typo, then I can reiterate the issue. I still want it solved, but till then, thanks.
 
Another is for x! is, "Omit even numbers, x is divisible by 3".

omit even means only odd numbers: 1,3,5,7,9,11,13,15,17
but only divisible by three means that 1,5,7,11,13,17... are not allowed; which leaves: 3, 9, 15, ... which is 3(1,3,5,...) ...

i.e. it is another way of asking for the product of odd-integer multiples of three.
So-

If x is an odd-integer multiple of 3, then the product of odd-integer multiples of 3 up to x is given by:[tex]3\prod_{n=1}^{x/3}(2n-1) = 3\left ( \frac{x}{3} \right )![/tex]... isn't it?

This sort of word problem is an exercise in curly thinking.
 
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