[Math] How many bit strings are there containing # 0’s, and # 1’s… Counting Problem Help

combinatoricsdiscrete mathematics

I'm struggling on these 2 problems:

  • How many 16 bit strings are there containing six 0's and ten 1's
    with no consecutive 0's?

  • How many 8 bit strings with exactly 2 1's are there such that the 1's
    are not adjacent?

For the first one, I am guessing they must alternate between 0 and 1 but I have no clue how to set this up. How can I use combinations to find this?

Best Answer

Don't space out; select the spaces!   Put 0,0,0,0,0, and 0 into _1_1_1_1_1_1_1_1_1_1_

You have ten 1 and eleven spaces to place the six 0 such that only one 0 goes in any space. How many ways can you select six of these eleven spaces?

Second verse, same as the first. (Only the numbers are changed.)

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