Answer: 7 Choose 4 | Solve Combination Question

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The discussion focuses on calculating the number of sequences of zeros and ones of length 7 that contain exactly 4 ones and 3 zeros, which is represented as 7 choose 4 (7C4). The reasoning is that by selecting any four spaces from the seven to place the ones, the remaining spaces will automatically be filled with zeros. The identity 7C4 equals 7C3 is highlighted, illustrating that choosing 4 positions for ones is equivalent to choosing 3 positions for zeros. Participants emphasize understanding the combinatorial basis through rearranging a specific sequence of digits. This approach helps clarify the application of the "N choose R" formula in solving the problem.
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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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