GATE CSE 1999 | Question: 1.3

Question

The number of binary strings of $n$ zeros and $k$ ones in which no two ones are adjacent is

• A. $^{n-1}C_k$
• B. $^nC_k$
• C. $^nC_{k+1}$
• D. None of the above

Comments

but that's the actual gate question mentioned with the condition so I asked him to edit and if the mention that condition the possibility we have to consider all the possibilities for n=k-1,k,k+1..... and that's what i wrote above anyways  for both we get none of the above ...they didn't mention n as a fixed value in the question if u don't care about what they gave in the book u can see from the original gate paper of 1999.....but i dont think its available anywhere but if they donot mention that condition we dont need to do  OR for all the possible cases.. i

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Please can anyone explain why only case of n distinct zeroes being considered? we could also have a case like, 10010001. In that case C(n+1, k) wont be true. Please comment

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Distinct doesn't mean that the zeroes can’t be adjacent to each other. We just need that no two ones are together.

Answer 1

 

Since there are n zeroes, so
XOXOXOXOXOXOX
n+1 gaps can be possible, where 1's can be placed so that no two one's are adjacent. So, no. of ways in which k 1's can be placed in n+1 gaps are,
$^{n+1}$$C_k$

Hence Option “D” None of these is the answer