Count the number of binary representations that have the given number of occurrences.

Question

How many binary representation exists that each has five "00", three "01", three "10" and three "11" in it ?!

Answer 1

See in the given problem that you can only have 3 '01' and 3 '10' meaning that you have to change the sequence of a binary string 3 times. For example in the binary represantation 11111011101101 I have changed the string a total of 4 times and got the abovementioned requirments. Now we can view this problem as more of a stars and bars problem. We can distribute the 5 '00's into the three strings of zeroes ie which would give us these combinations for strings:(A string signifies a series of a single number for example 111111111 is a string of ones)

{1,1,3} - 3 cases

{2,2,1} - 3 cases

{0,1,4} - 6 cases

{0,0,5} - 3 cases

{3,2,0} - 6 cases

Now similarly for 1 we have 4 strings and 3 '11's.

{0,2,1,0} - 12 cases (0 signifies there is only one '1' in that string and the pair does not form)

{1,1,1,0} - 4 cases

{0,0,3,0} - 4 cases

Notice that each case case gives a unique representation

Now we just have to multiply the 0 cases by 1 cases and add them all

$3*20+3*20+3*20+6*20+6*20$ =420

Hope this helps:)