Heres a binary word of length 12 001011010101 As you go from

Here\'s a binary word of length 12: 001011010101. As you go from the first 0 down to the last 1, you can count a total of 6 0\'s and 6 l\'s, and as you count, the number of 0\'s is never less than the number of 1 \'s. How many binary words of length 12 have that property? ANS.:_______________ Show the \"A number\" of the Catalan sequence in OEIS:____________ Figure out all of Catalan numbers from b_0 up to b_12. ANS.:________________________________________ ______________________________________________ How many upright paths from (0, 0) to (6, 6) never cross the line y = x? (\"Upright path\" is defined in problem 3a.) ANS.:_______________________________

Solution

to find haw may binary word which have property that

total count of o\'s and 1\'s in b₁₂ is same

that there is 6 0\'s and 6 1\'s

since b₁₂ there is 12 positions of 0 or 1\'s

since number of 0\'s=number of 1\'s

we have 6 position which have each 2 choices

that is for 0\'s there are 6 position for 0\'s

each position have 2 choices

therefore choice of number of 6 0\' and 6 1\'s =2x2x2x2x2x2

=64

therfore there are 64 binary numbers which have given properties