Probability that 2 heads do not come consecutively. 
A fair coin is tossed $10$ times. Then the probability that two heads do not appear consecutively is?

Attempt:
Cases are:
Given condition cannot be met with $10, 9, 8, 7$ or $6$ heads.
1) 5 heads + 5 tails.
First fulfilling the essential condition we get:
$\mathrm{HTHTHTHTH}$
Then we'll be left with 1 tail which can be placed at 6 places.
So number of cases = $6$
2) $4$ heads, $6$ tails
$\mathrm{HTHTHTH}$
Left with $3$ tails.
So all $3$ can be placed together at $5$ places.
Or $3 = 2+1$, in which case we have $5 \times 4 = 20$ cases.
Total number for #2 $= 20 + 5 = 25$
3) 3 heads, 7 tails.
$\mathrm{HTHTH}$
Left with $5$ tails.
All 5 can be placed together, so in that case we have $4$ cases.
Or $5 = 2+3= 1+4$
For $5 = 2+3 = 1+4$, we have $4\times 3$ cases each.
Therefore total number of cases for #3 $= 12+12+ 4 = 28$
4) 2 heads 8 tails.
$\mathrm{HTH}$
$7$ left
$7 = 7+ 0$ , $3$ cases.
$7 = 6+1= 5+2= 3+4$ , $6$ cases each
So number of cases for #4 = $6\times 3 + 3 = 21$
5) 1 head 9 tail, 10 cases
6) 0 heads, 1 case.
So total number of cases = $6+25+28+21+10 +1  = 91$
I tried thrice and got $91$ cases only.
So answer should be $P(E) = \dfrac{91}{2^{10}}$
But that's not the right answer.
Please tell me my mistake.

Given that a fair coin is tossed $10$ times. So, the possible outcomes are $2^{10}$ 
Now to find the probability that no two heads appear consecutively, then the possible outcomes are 
$$\begin{matrix} Heads & Tails & Total \\ 0 & 10 & \dbinom{11}{0} \\ 1 & 9 & \dbinom{10}{1} \\ 2 & 8 & \dbinom{9}{2} \\ 3 & 7 & \dbinom{8}{3} \\ 4 & 6 & \dbinom{7}{4} \\ 5 & 5 & \dbinom{6}{5} \\  \end{matrix}$$
We shouldn't consider the cases where heads are $>6$, otherwise there will be two heads consecutively. So, we neglect those cases.
Since, all the outcomes are mutually exclusive$$\left(1+\dbinom{10}{1}+\dbinom{9}{2}+\dbinom{8}{3}+\dbinom{7}{4}+\dbinom{6}{5}\right)\left(\dfrac{1}{2^{10}}\right)$$
$$(1+10+36+56+35+6)\left(\dfrac{1}{2^{10}}\right)=\dfrac{144}{2^{10}}$$
