how many bit strings of length 10 have a exactly three 0s b

how many bit strings of length 10 have

a) exactly three 0s?

b) more 0s than 1s?

c) at least seven 1s?

d) at least three 1s?

Solution

The number of bit strings of length 10 with n 0\'s (or n 1\'s in fact):
is C(10,n) , where C(a,b) = a! / [(a-b)!b!] is the combinitorial function.

So the first answer is C(10,3) = 120

In the second, there could be 6,7,8,9, or 10 zeros, so:
C(10,6) + C(10,7) + C(10,8) + C(10,9) + C(10,10)
= 210 + 120 + 45 + 10 + 1 = 386

In the third, C(10,7) + C(10,8) + C(10,9) + C(10,10)
120 + 45 + 10 + 1 = 176

In the fourth,
C(10,3) + C(10,4) + C(10,5) + C(10,6) + C(10,7) + C(10,8) + C(10,9) + C(10,10)
= 2^10 - C(10,2) - C(10,1) - C(10,0)
= 1024 - 45 - 10 - 1
= 968