1 how many 16 bits strings are there contaning six 0s and te

1) how many 16 bits strings are there contaning six 0\'s and ten 1\'s with no consecutive 0\'s?

2) how many 8 bits string with extaclty two 1\'s are there such that the 1st are not adjacent ?

Solution

1)

For the 0\'s to be non-adjacent,there must be 1\'s in between each of the 0\'s

The pattern has to start with 01010101010

Now, we can now put the remainder of the ones in any one of 7 positions.

This is done in ^5+7-1C_7-1 = 462 ways

2) For this part, we\'ll answer how many 8 bits strings contain exactly two 1\'s

= ⁸C₂ = 28 bits

Now, consider all the cases where both the ones appear together.

There would be 7! ways to arrange 7 objects, but since 6 objects are identical (the 0\'s) we divide by 6!

Number of ways where the 1\'s appear together = 7! / 6! = 7ways

Thus, number of ways where the 1\'s don\'t appear together are :

28 - 7 = 21 ways

Hope this helps. Ask if you have doubts.