help! permutations and combinations?

Question

1. A binary string is an arrangement of 0's and 1's. A binary string of the length of 8 is a byte.

a) how many different bytes are there?
b) how many different bytes are there with five 0's three 1's and no two consecutive 1's.








2. in mrs. smith's 14-member music class, there is a group of 5 students who are disruptive whenever any two of them are next to each other. in how many ways can mrs. smith line up the students to sing a song so that no two of the five disruptive students are next to each other.

Answer 1

1a) 2^8
1b) (4+3+2+1) + (3+2+1) + (2+1) + 1 = 20
10101000
10100100
10100010
10100001
10010100
10010010
10010001
10001010
10001001
10000101

01010100
01010010
01010001
01001010
01001001
01000101

00101010
00101001
00100101

00010101

2)
If n = number of students
k = number of troublemakers
Then the number of ways to keep the troublemakers separated is:
(C(n-k,k) + C(n-k,k-1)) * k! * (n-k)!
= C(n-k+1, k) * k! * (n-k)!

In this case
n= 14, k = 5 so we have
C(10, 5) * 5! * 9!
= 10! * 9! / 5!

By the way, the first part of the expression, C(n-k+1, k) corresponds to what you are looking for in part 1b, where you are not distinguishing between the troublemakers and the non trouble makers:
There n=8, k=3 so
C(n-k+1,k) = C(6, 3) = 20

Note to self: problem boils down to selecting k integers from n such that no two selected integers are adjacent (differ by 1). So, want to select contiguous blocks length 1 or more. So each selection peels off one element (a 0) from the block and one for the block separator (a 1). This brings us down to n-k. The second term is since the first selector (the 1) might have a block of zero 0s before it.

Answer 2

a0 there are two possibilities of filling each of the 8 places. So de desired number of different bytes =2^8 answer
b0

Answer 3

1.a 2^8
1.b 8C5-(6 ways you can have three ones in a row) - ( 5C1+4C1+4C1+4C1+4C1+4C1+5C1) This last part represents the number of ways you can have two 1's in a row.
2. Ran out of time for this one--sorry.