View source for 1. How many bitstrings of length 10 have exactly four zeros? - Rhea

1. How many bitstrings of length 10 have exactly 4 zeros?

We know that every bitstring has exactly 4 zeros, meaning that it also has exactly 6 ones.  So, I think that the solution to this problem would be to find the number of ways that 4 zeros can be placed into 10 spaces and then 6 ones placed into the remaining 6 spaces.

So, placing 4 indistinguishable objects (zeros) into 10 spaces:

C(10,4)

And, placing 6 indistinguishable objects (ones) into the remaining 6 spaces:

C(6,6)

Gives us

C(10,4)*C(6,6) = (10!/(4!*6!))*(6!/(0!*6!)) = (7*8*9*10)/(1*2*3*4) * 1 = 5040/24 = 210