Question

How many bit-strings are there of length 128 that either start with 1111 or end with 0000 but not both?

Answer 1

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Answer:

Strings starting with 1111:

Let the first 4 digits be 1111. The rest of 124 digits can be 0 or 1

=> We have a permutation of n^r => 2¹²⁴

Strings ending with 0000:

Let the last 4 digits be 0000. The first 124 digits can be 0 or 1

=> We have a permutation of 2¹²⁴

Strings starting with 1111 and ending with 0000

If we have first 4 digits as 1111 and last 4 digirs as 0000, then the rest of 120 digits can be 0 or 1

=>  => We have a permutation of 2¹²⁰

Total  bit-strings are there of length 128 that either start with 1111 or end with 0000 but not both

=> 2 * 2¹²⁴ - 2¹²⁰

= 2¹²⁵ - 2¹²⁰

= 2¹²⁰ ( 2⁵ - 1)