Combinations Question

C

Chlo

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Mar 30, 2015
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Hi I'm having trouble with the following question:

Q: How many bit strings of length 10 contain at most three 1s AND at least seven zeros?

So this is what I did:

First find the number of bit strings of length 10 which is 2^10= 1024

Then find the number of bit strings of length 10 with at least seven zeros which is:
1024 - (10 choose 0)- (10 choose 1)- (10 choose 2)- (10 choose 3)- (10 choose 4)- (10 choose 5)- (10 choose 6) = 176

Then find the number of bit strings of length 10 with at most 3 zeros which is:
(10 choose 0)+(10 choose 1)+(10 choose 2)+(10 choose 3)= 176

I'm having trouble figuring out what to do next. Do I add these two calculated numbers (176+176)? Or is it something else?
 
Chlo said:
Hi I'm having trouble with the following question:
Q: How many bit strings of length 10 contain at most three 1s AND at least seven zeros?
Click to expand...
\(\displaystyle \displaystyle{\sum\limits_{k = 0}^3 {\binom{10}{k}}} \)
 
Chlo said:
So it's just (10 choose 0) + (10 choose 1) + (10 choose 2) + (10 choose 3) ?
Click to expand...
Correct! It in just the number of subsets of a set of ten which have at most three elements.
You need to convince yourself why that is the case. Hint: put a 1 in each selected space.
 
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