Bankers Algorithm-Is the system in safe state? Could somebody please provide a step-through approach to solving the following problem using the Banker's Algorithm?

A system contains 10 units of resource class Ru. The resource requirements of three user processes P1, P2 and P3 are as follows

                      |   P1  |  P2  |   P3   |
Max Requirement       |   8   |  7   |   5    |
Current Allocation    |   3   |  1   |   3    |
Balance Requirement   |   5   |  6   |   2    |
New Request Made      |   1   |  0   |   0    |


Using Banker’s algorithm, determine if the projected allocation state is safe and whether the request of P1 will be granted or not.

My solution:

total recources :10
total allocation :7
available: 10 -7 = 3

Need 
P1     5
P2     6
P3     2

3&gt;2 so P3 can run now available is 6
6&gt;5 so P1 can run now availabl 9
9&gt;6 so P2 can run 


System is safe state,Hence P1 will be granted.

But I'm not sure if my solution is true.
you can say there is no deadlock if there is a safe sequence for the completion, after $P1$ request is granted there must be a safe sequence where by which every process is completed.

After P1 request is granted
                      |   P1  |  P2  |   P3   |
Max Requirement       |   8   |  7   |   5    |
Current Allocation    |   4   |  1   |   3    |
Balance Requirement   |   4   |  6   |   2    |
now the resources left over are 10-4-1-3=2


so can we complete with 2 resources? any sequence is possible?

YES

if P3 is given 2 resources it will complete and release all the current allocated resources so our resource count becomes 5 , now we have 5 and we can complete P1 and get back 4 now our resource count is 9 and i need just 6 to complete P2 so a safe sequence is possible.
so safe sequence will be 

$P3 \rightarrow P1\rightarrow P2$ 

so $P1$ request is granted