Answer: 7 Choose 4 | Solve Combination Question

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Discussion Overview

The discussion revolves around understanding the combinatorial problem of determining how many sequences of zeros and ones of length 7 contain exactly 4 ones and 3 zeros. Participants explore the reasoning behind the use of the binomial coefficient "7 choose 4" in this context.

Discussion Character

  • Exploratory
  • Mathematical reasoning

Main Points Raised

  • One participant questions why the number of sequences is represented as "7 choose 4".
  • Another participant explains that selecting any four spaces from seven to place the ones leads to the conclusion that the remaining spaces will automatically be filled with zeros.
  • Some participants note the identity that "7 choose 4" is equal to "7 choose 3", suggesting a symmetry in the selection of ones and zeros.
  • A later reply suggests starting with a specific arrangement of digits (1111000) and finding all unique rearrangements to develop an understanding of the factorial relationship involved in the "N choose R" formula.

Areas of Agreement / Disagreement

Participants generally agree on the reasoning behind using "7 choose 4" and the identity relating "7 choose 4" to "7 choose 3". However, there is no consensus on the best approach to intuitively understand the problem, as different methods are suggested.

Contextual Notes

The discussion does not resolve the deeper intuition behind the combinatorial reasoning, and participants present various methods without establishing a definitive understanding.

Who May Find This Useful

Individuals interested in combinatorial mathematics, particularly those seeking to understand the application of binomial coefficients in counting problems.

Bachelier
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How many sequences of zeros and ones of length 7 contain exactly 4 ones and 3 zeros?

I'm having a hard time figuring out why it is 7 Chooses 4?

thx
 
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Select any four spaces within the seven spaces in 7C4 ways, and put 1's in each of the four spaces. Where can all the zeros go now? Notice the identity 7C4=7C(7-4)=7C3. Can
you see, in the context of this problem , why 7C4=7C3?
 
Bacle said:
Can
you see, in the context of this problem , why 7C4=7C3?

Yes, because we for 3 zeros and 4 ones.
 
Exactly. Perfect. This is the general identity nCk =nC(n-k).
 
Bacle said:
Exactly. Perfect. This is the general identity nCk =nC(n-k).

Thank you :)
 
No problem; glad to help.
 
Bachelier said:
How many sequences of zeros and ones of length 7 contain exactly 4 ones and 3 zeros?

I'm having a hard time figuring out why it is 7 Chooses 4?

thx

If you want to get some intuition, what you can do is to start with 1111000, and then find every way of re-arranging these digits so that each rearrangement is different from each other. For example 1111000 goes to 1110100, 1110010, 1110001 and so on.

Using this as your intuition base, you can develop a factorial relationship which will eventually give you the "N choose R" formula that you use.
 

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