Pascal triangle and number 11

When I was playing with number 11, I find a rather familiar number pattern for 11^x
we all know that 11^2 = 121 and 11^3 = 1331
121 and 1331 remind me of the polynomial (a+b)^x which leads me to pascal triangle.



Look at the number rows above. The first represent 11^0 which is 1, the second represent 11^1 which is 11 and soon on.
does this mean that 11^5 = 15101051? of course not. It should be calculated this way.


So, the answer is 161051

This method could be applied for 11^x for any value of x due to the nature of no.11

11 = 10+1

Just replace the x with 10 and the y with 1, we will get



since will always be one,




notice that can be derived from pascal triangle.

The last digit of 11^x will always be 1 when the second last will always be the last digit of x

edit: I just realised that the long equation is being cut. Since it does not affect that much, I'll just leave it that way.